Testing a New Immune Cell Therapy, GD2-Targeted Modified T-cells (GD2CART), in Children, Adolescents, and Young Adults With Relapsed/Refractory Osteosarcoma and Neuroblastoma, The GD2-CAR PERSIST Trial

Inclusion Criteria: * Must have histologically confirmed neuroblastoma or osteosarcoma that is recurrent or refractory and for which standard curative measures do not exist or are no longer effective. Must have histologic verification of their disease at diagnosis or at relapse * Patients with osteosarcoma in the dose escalation cohort, must have evaluable or measurable disease at enrollment * Patients with osteosarcoma in the expansion cohort must have measurable disease by Response Evaluation Criteria in Solid Tumors version 1.1 (RECIST v1.1) at enrollment * Patients with neuroblastoma in the dose escalation or dose the expansion cohort must have: * Prior progressive disease OR refractory disease present since diagnosis AND at least one of the following: * Evidence of tumor in the bone marrow (BM) * At least one metaiodobenzylguanidine (MIBG)-avid soft tissue or skeletal site * For MIBG-nonavid disease, at least one FDG-PET- positive soft tissue or skeletal site plus past histologic confirmation * Progressive disease is defined as any disease progression occurring at any time after the diagnosis of high-risk neuroblastoma. Refractory disease is defined as an incomplete response of high-risk neuroblastoma to all treatments but without disease progression * Must be \< 40 years of age * There is no limit to the number of prior treatment regimens. The following washout periods prior to leukapheresis apply to patients undergoing leukapheresis on this study. If a patient has cryopreserved peripheral blood mononuclear cells (PBMCs) stored, the following washout periods are strongly recommended but not required and the product is useable if it meets the criteria established in this Investigational New Drug (IND) * Myelosuppressive chemotherapy: Patients must not have received myelosuppressive chemotherapy within 3 weeks of leukapheresis (6 weeks if prior nitrosourea) * Hematopoietic growth factors: At least 7 days must have elapsed since the completion of therapy with a short-acting growth factor. At least 14 days must have elapsed after receiving pegfilgrastim * Biological agent, tyrosine kinase inhibitor, targeted agent, metronomic chemotherapy: At least 7 days must have elapsed since the completion of therapy with a biologic agent, tyrosine kinase inhibitor, targeted agent, or metronomic non-myelosuppressive regimen * 131I-MIBG or other radioisotope therapy: At least 6 weeks must have elapsed since prior therapy with 131I-MIBG. At least 6 weeks or 10 half-lives (whichever is shorter) must have elapsed since prior therapy with any other radioisotope * Monoclonal antibodies and checkpoint inhibitors: At least 3 weeks or 5 half-lives (whichever is shorter) must have elapsed since prior therapy that included a monoclonal antibody or checkpoint inhibitor * Radiotherapy (XRT): 3 weeks must have elapsed since XRT; at least 6 weeks if XRT involved central nervous system (CNS) or lung fields; and at least 12 weeks from total-body irradiation (TBI), craniospinal XRT, or XRT involving ≥ 50% bony pelvis, with the exception that there is no time restriction for palliative radiation with minimal bone marrow involvement and the patient has measurable/evaluable disease outside the radiation port or the site of radiation has documented progression * Vaccine therapy, anti-GD2 mAb therapy, or therapy with any genetically engineered T cells: Patients may have received previous vaccine therapy, anti-GD2 monoclonal antibody (mAb) therapy, or therapy with any genetically engineered T cells except prior GD2 CAR T cell therapy. At least 3 weeks or 5 half-lives, whichever is shorter, must have elapsed since any prior vaccine or monoclonal antibody therapy. At least 42 days must have elapsed since prior modified T cell, natural killer (NK) cell, or dendritic cell therapy * Allogeneic stem cell transplant/infusion: At least 12 weeks must have elapsed since allogeneic stem cell transplant and without evidence of active graft versus host disease (GVHD) * Autologous stem cell transplant/infusion: Patients who received an autologous stem cell infusion following myeloablative therapy should be at least 6 weeks from their infusion. Patients who received an autologous stem cell infusion following non-myeloablative therapy do not have a wash-out period; they are eligible once they meet all other eligibility requirements, including recovery from acute side effects * Must meet parameters for apheresis per institutional guidelines. (This criterion does not apply to patients with apheresis product or usable T cell product available for use). Cryopreserved PBMCs stored from participation in other institutional cell therapy or cell collection studies or standard of care may be used to generate the cellular product on this study if they meet the criteria established in this IND * Patients \> 16 years of age must have Karnofsky ≥ 60%. Patients ≤ 16 years of age must have Lansky scale ≥ 60%; corresponding to Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 * Leukocytes \>= 750/mcL (For patients without bone marrow involvement. Patients who have bone marrow involvement with tumor are exempt from the platelet requirement and will not be evaluable for hematological toxicities. Patients must not be refractory to transfusions) * Absolute neutrophil count (ANC) ≥ 500/mcL (For patients without bone marrow involvement. Patients who have bone marrow involvement with tumor are exempt from the platelet requirement and will not be evaluable for hematological toxicities. Patients must not be refractory to transfusions.) * Platelets \>= 75,000/mcL (transfusion independent defined as no transfusion in prior 7 days) (For patients without bone marrow involvement. Patients who have bone marrow involvement with tumor are exempt from the platelet requirement and will not be evaluable for hematological toxicities. Patients must not be refractory to transfusions.) * Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT)(serum glutamate pyruvate transaminase \[SGPT\]) =\< 5 x upper limit of normal (ULN) (For the purpose of this study, the ULN for SGOT is 50 U/L and the ULN for SGPT is 45 U/L) * Albumin \>= 2 g/dL * Total bilirubin =\< 2 x institutional upper limit of normal (ULN) for age. Patients with Gilbert's syndrome are excluded from the requirement of a normal bilirubin and patients will not be excluded if bilirubin elevation is due to tumor involvement. (Gilbert's syndrome is found in 3-10% of the general population, and is characterized by mild, chronic unconjugated hyperbilirubinemia in the absence of liver disease or overt hemolysis). Note: Adult values will be used for calculating hepatic toxicity and determining eligibility * Age, maximum serum creatinine (mg/dL): * 1 month to \< 6 months: 0.4 (male), 0.4 (female) * 6 months to \< 1 year: 0.5 (male), 0.5 (female) * 1 to \< 2 years: 0.6 (male), 0.6 (female) * 2 to \< 6 years: 0.8 (male), 0.8 (female) * 6 to \< 10 years: 1 (male), 1 (female) * 10 to \< 13 years: 1.2 (male), 1.2 (female) * 13 to \< 16 years: 1.5 (male), 1.2 (female) * \>= 16 years: 1.7 (male), 1.4 (female) OR Creatinine clearance or glomerular filtration rate (GFR) \>= 60 mL/min/1.73 m\^2 for patients with levels above institutional normal * Cardiac ejection fraction \>= 45% or shortening fraction \>= 28%, no evidence of physiologically significant pericardial effusion as determined by an echocardiogram (ECHO). No clinically significant electrocardiogram (ECG) findings * Pulmonary status: No clinically significant pleural effusion. Baseline oxygen saturation \> 92% on room air at rest * No acute neurotoxicity greater than grade 2 with the exception of decreased tendon reflex (DTR). Any grade of DTR is eligible * Females of child-bearing potential and males of reproductive potential who are sexually active must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) from the start of study enrollment until 6 months after GD2 CAR T cell infusion (or until the duration of study participation, in the case of patients who start lymphodepleting chemotherapy but do not receive the GD2 CART infusion). Should a female become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately * Note: Females of childbearing potential are defined as those who are past the onset of menarche and are not surgically sterile (i.e., bilateral salpingectomy, bilateral oophorectomy, complete hysterectomy) or post menopausal * All patients \>= 18 years of age must be able to give informed consent or if unable to give consent have a legal authorized representative (LAR) who can give consent for the patient. For patients \< 18 years old their LAR (i.e., parent or legal guardian) must give informed consent. Pediatric patients will be included in age appropriate discussion and verbal assent will be obtained for those \> 7 years of age, when appropriate, according to local policy Exclusion Criteria: * Receiving any other current investigational agents * History of anaphylactic reactions attributed to anti-GD2 antibodies or to compounds of similar chemical or biologic composition to GD2CART, cyclophosphamide, fludarabine, or other agents used in this study. History of hypersensitivity to dornase alfa, Chinese hamster ovary cell products, or any of the components of pulmozyme * Patients who require systemic corticosteroid or other immunosuppressive therapy. (A one-week washout from systemic corticosteroid or other immunosuppressive therapy is permitted.) Use of physiologic doses of corticosteroids (up to 3 mg/m\^2/day prednisone equivalent) are permitted. Use of topical, ocular, intra-articular, intra-nasal, or inhaled corticosteroids are permitted * Uncontrolled intercurrent illness including, but not limited to, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * History of additional malignancy other than non-melanoma skin cancer or carcinoma in situ (e.g., cervix, bladder, breast) unless untreated and stable or disease free for at least 3 years * Untreated central nervous system (CNS) metastasis. Patients with previous CNS tumor involvement that has been treated and is stable for at least 6 weeks following completion of therapy are permitted. Patients who are clinically stable as evidenced by no requirements for corticosteroids, no evolving neurologic deficits, and no progression of residual brain abnormalities without specific therapy, are permitted. Patients with asymptomatic subcentemeric CNS lesions are permitted if no immediate radiation or surgery is indicated * CNS disorder such as cerebrovascular ischemia/hemorrhage, dementia, cerebellar disease, or autoimmune disease with CNS involvement that in the judgement of the investigator may impair the ability to evaluate neurotoxicity * Presence of fungal, bacterial, viral, or other infection that is uncontrolled * Ongoing infection with human immunodeficiency virus (HIV), hepatitis B (hepatitis B surface antigen \[HBsAg\] positive), or hepatitis C virus (anti-HCV positive) as the immunosuppression contained in this study will pose unacceptable risk. A history of HIV, hepatitis B, or hepatitis C is permitted if the viral load is undetectable per quantitative polymerase chain reaction (PCR) and/or nucleic acid testing * Primary immunodeficiency or history of systemic autoimmune disease (e.g., Crohns, rheumatoid arthritis, systemic lupus) requiring systemic immunosuppression/systemic disease modifying agents within the last 2 years * Females of childbearing potential must have a negative serum or urine pregnancy test. Pregnant females are excluded from this study because the effects of autologous GD2CART on the developing human fetus are unknown and because the chemotherapy agents used in this trial (cyclophosphamide and fludarabine) are category D agents with the potential for teratogenic or abortifacient effects. Additionally, because there is an unknown but potential risk for adverse events (AEs) in nursing infants secondary to treatment of the mother with cyclophosphamide/fludarabine, breastfeeding should be discontinued if the mother is treated with cyclophosphamide/fludarabine. These potential risks may also apply to other agents used in this study * In the investigator's judgment, unlikely to complete protocol-required study visits or procedures, including follow-up visits, or comply with the study requirements for participation. Or in the investigator's judgment, if the patient is likely to develop significant toxicity and morbidity from CAR-T cell expansion mediated inflammation based on location of tumor site