Necitumumab and Trastuzumab in Combination With Osimertinib for the Treatment of Refractory Epidermal Growth Factor Receptor (EGFR)-Mutated Stage IV Non-small Cell Lung Cancer

Inclusion Criteria: * Willing and able to provide informed consent * Cytologically or histologically confirmed non-small cell lung cancer (NSCLC), that is stage IV (metastatic), with an activating and sensitizing EGFR mutation (e.g., exon 20 insertion mutations are excluded). Enrollment of patients with mutations other than exon 19 deletion and the L858R point mutation require literature supporting sensitivity to EGFR tyrosine kinase inhibitors * Progressed on osimertinib. Osimertinib must have been included as the last systemic therapy prior to trial enrollment. This excludes patients who received osimertinib in combination with other EGFR-tyrosine kinase inhibitor (TKI) or anti-human epidermal growth factor receptor (anti-HER) therapy * Measurable disease, as per RECIST 1.1 * Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 * Able to swallow the study drugs, has no known intolerance of study drugs or excipients, and able to comply with study requirements * Absolute neutrophil count (ANC) \>= 1,500/mcL * Platelets \>= 100,000/mcL * Hemoglobin \>= 9 g/dL or \>= 5.6 mmol/L * Serum creatinine =\< 1.5 X upper limit of normal (ULN) OR measured or calculated creatinine clearance \>= 50 mL/min for participants with creatinine levels \> 1.5 X institutional ULN (Glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinine clearance \[CrCl\]) * Creatinine clearance may be calculated following institutional practices * Serum total bilirubin =\< 1.5 X ULN OR direct bilirubin =\< ULN. * If a patient experiences elevated alanine aminotransferase (ALT) \> 5 X ULN and elevated total bilirubin \> 2 X ULN, clinical and laboratory monitoring should be initiated by the investigator. For patients entering the study with ALT \> 3 X ULN, monitoring should be triggered at ALT \> 2 X baseline * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) and ALT (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X ULN OR =\< 5 X ULN for participants with liver metastases. * If a patient experiences elevated ALT \> 5 X ULN and elevated total bilirubin \> 2 X ULN, clinical and laboratory monitoring should be initiated by the investigator. For patients entering the study with ALT \> 3 X ULN, monitoring should be triggered at ALT \> 2 X baseline * Albumin \>= 2.5 g/dL. * International Normalized Ratio (INR) or Prothrombin Time (PT), and activated partial thromboplastin Time (aPTT) =\< 1.5 X ULN unless participant is receiving anticoagulant therapy, and then only as long as PT or PTT is within therapeutic range of intended use of anticoagulants * Left ventricular ejection fraction \>= 50% as measured by transthoracic or transesophageal echocardiogram within 60 days prior to receiving study treatment * Female participants of childbearing potential must have a negative urine or serum pregnancy test. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. The serum pregnancy test must be negative for the participant to be eligible, and the participant must agree to use two highly-effective methods of birth control from the time of the first study drug treatment through 180 days after the last study drug treatment, or be of non-childbearing potential. Non-childbearing potential is defined as follows (by other than medical reasons): * \>= 45 years of age and has not had menses for \> 2 years, * Participants who have been amenorrhoeic for \< 2 years without history of a hysterectomy and oophorectomy must have a follicle stimulating hormone value in the postmenopausal range upon screening evaluation, or * Post-hysterectomy, post-bilateral oophorectomy, or post-tubal ligation * Male participants must use a condom when having sex with a pregnant woman and when having sex with a woman of childbearing potential from the time of the first study-drug treatment through 180 days after the last study drug treatment. Contraception should be considered for a non-pregnant female partner of childbearing potential * Male and female participants must agree not to donate sperm or eggs, respectively, from the first study-drug treatment through 180 days after the last study drug treatment * Female participants must agree to not breastfeed during the study or for 180 days after the last dose of study treatment * Participants must agree to not donate blood during the study or for 90 days after the last dose of study treatment Exclusion Criteria: * Concurrent enrollment in another clinical study, unless enrolled only in the follow-up period or an observational study * Any chemotherapy, immunotherapy, biologic, or hormonal therapy for cancer treatment in the prior 3 weeks, other than osimertinib. Concurrent use of hormones for non-cancer-related conditions (eg, insulin for diabetes and hormone replacement therapy) is acceptable. Stereotactic, palliative radiation for symptomatic bone metastases is acceptable without a washout. Stereotactic brain radiation for asymptomatic brain metastases is acceptable with a 7 day washout * Use of any investigational anticancer therapy received within 28 days prior to the first dose of study drugs * Prior treatment with the combination of two or more of the following therapies which target EGFR/HER: osimertinib (Tagrisso), erlotinib (Tarceva), gefitinib (Iressa), afatinib (Gilotrif), lapatinib (Tykerb), neratinib (Nerlynx), vandetanib (Caprelsa), cetuximab (Erbitux), trastuzumab (Herceptin), pertuzumab (Perjeta), ado-trastuzumab emtansine (Kadcyla), panitumumab (Vectibix), necitumumab (Portrazza), dacomitinib (Vizimpro), poziotinib, or other combination deemed as "combined HER-inhibition therapy" by the investigator * Has not recovered (recovery is defined as National Cancer Institute \[NC\] Common Terminology Criteria for Adverse Events \[CTCAE v5.0\] grade =\< 1) from the acute toxicities of previous therapy, except treatment-related alopecia, sensory neuropathy, or laboratory abnormalities otherwise meeting the inclusion requirements stated in the inclusion criterion. Other grade 2 or less toxicities not constituting a safety risk based on the investigator's judgment are acceptable * Known small cell lung cancer or small cell transformation * The patient has a known allergy / history of hypersensitivity reaction to any of the treatment components or any other contraindication to one of the administered treatments * History or evidence of current clinically relevant coronary artery disease \>= grade III by the Canadian Cardiovascular Society Angina Grading Scale or uncontrolled congestive heart failure of current \>= class III as defined by the New York Heart Association, or unstable cardiac arrhythmia requiring medication (atrial fibrillation is permitted if clinically stable) * The patient has experienced myocardial infarction within 6 months prior to study enrollment * The patient has any ongoing or active infection, including active tuberculosis or known infection with the human immunodeficiency virus, or active, clinically significant infection requiring the use of parenteral anti-microbial agents, or grade \> 2 by NCI CTCAE (v5.0) within 14 days prior to enrollment * The patient is pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the screening visit through 180 days after the last dose of trial treatment * Progressive or symptomatic brain metastases. Brain metastases that have been radiated, are asymptomatic, and on a stable or decreasing dose of steroids are allowed. Leptomeningeal disease is excluded * Serious accompanying disorder or impaired organ function, including major surgery within 3 weeks before randomization * Requirement for IV alimentation (at the time of enrollment) * History of another cancer within 2 years of study initiation, with the exception of fully treated cancers unlikely to affect the assessment of the study treatment safety or efficacy including early stage breast, prostate, non-melanomatous skin, thyroid, cervical, or endometrial cancer * Gastrointestinal disorder affecting absorption * Participants must not be considered a poor medical risk due to a serious, uncontrolled medical disorder, nonmalignant systemic disease, or active, uncontrolled infection. Examples include, but are not limited to, uncontrolled ventricular arrhythmia, uncontrolled major seizure disorder, unstable spinal cord compression, superior vena cava syndrome, or any psychiatric disorder that prohibits obtaining informed consent * Patient unwilling or unable to comply with the protocol * Any condition that, in the opinion of the investigator or sponsor, would interfere with evaluation of the investigational product or interpretation of subject safety or study results * History of arterial or venous thromboembolism within 3 months prior to study enrollment. Patients with a history of venous thromboembolism beyond 3 months prior to study enrollment can be enrolled if they are appropriately anticoagulated * Recent (within 30 days before enrollment) or concurrent yellow fever vaccination