A Pilot Study of Immunotherapy Including Haploidentical NK Cell Infusion Following CD133+ Positively-Selected Autologous Hematopoietic Stem Cells in Children With High Risk Solid Tumors or Lymphomas

The transplant recipient will be evaluated for eligibility at two time points during study participation. The first phase will be when the autologous stem cell product is collected. The recipient will later need to meet specific eligibility criterion at the time of the autologous stem cell infusion. The two phases and the respective criteria are described below. Inclusion criteria for autologous stem cell collection (Phase 1 - transplant recipient): * Less than or equal to 21 years of age. * Malignancy at high risk of treatment failure for which autologous hematopoietic stem cell transplantation is considered within standard practice. * Group A: High-risk neuroblastoma * Group B: Recurrent or refractory Hodgkin lymphoma; recurrent or refractory non-Hodgkin lymphoma * Group C: High-risk, recurrent or metastatic sarcoma; recurrent or advanced stage Wilms tumor; desmoplastic small round cell tumor; metastatic or recurrent retinoblastoma, high-risk germ cell tumors, and high-risk brain tumors * Sarcoma or Wilms tumor diagnosis (Group C) will require evaluation by physician in the St. Jude Solid Tumor Division, other than the referring physician, attesting that autologous SCT provides the prospect of direct benefit for the participant. * Has a potentially suitable human leukocyte antigen (HLA) haploidentical donor available. * Research participant or legal guardian/representative must be willing to give written informed consent * Does not have any active or prior malignant or pre-malignant condition of the bone marrow, excluding metastasis of the primary malignancy. * Has no known allergy to murine products or positive human anti-mouse antibody (HAMA). * (Female only) Negative serum or urine pregnancy test (to be conducted within 7 days prior to enrollment). * (Female only) Not breastfeeding. Inclusion criteria to proceed with autologous stem cell transplantation (Phase 2 - transplant recipient): * Has a confirmed suitable HLA haploidentical donor available. * Previously collected autologous stem cell product met the minimum collection target and minimum infusion target as described in the protocol. * At least two weeks since receipt of any biological therapy, chemotherapy, and/or radiation therapy. * Has recovered from all acute NCI Common Toxicity Criteria grade II-IV non-hematologic toxicities from prior therapy per the judgment of the PI. * Shortening fraction greater than or equal to 25%. * Creatinine clearance or glomerular filtration rate greater than or equal to 50 mL/min/1.73 m\^2. * Pulse oximetry greater than or equal to 92% on room air. * Alanine aminotransferase (ALT) and aspartate transaminase (AST) less than or equal to 3 times the upper limit of the institution-established normal range. * Direct bilirubin less than or equal to 3.0 mg/dL. * Karnofsky or Lansky performance score of greater than or equal to 50. * Has not received a prior hematopoietic stem cell transplant within 3 months. * Has no known allergy to murine products or positive human anti-mouse antibody (HAMA) * (Female only) Is not pregnant (negative serum or urine pregnancy test to be conducted within 7 days prior to admission for transplant). * (Female only) Is not breastfeeding. * Does not meet donation eligibility requirements as outlined by 21 CFR 1271 and agency guidance. Inclusion criteria for haploidentical NK cell donor: * At least 18 years of age. * Partially HLA matched family member. * Human immunodeficiency virus (HIV) negative. * (Female only) Is not pregnant (negative serum or urine pregnancy test to be conducted within 7 days prior to enrollment). * (Female only) Is not breastfeeding.