Targeted Treatment for Advanced Non-Small Cell Lung Cancer That Has Increased Copies of the MET Gene (An Expanded Lung-MAP Treatment Trial)

Inclusion Criteria: * Participants must have been assigned to S1900J by the Southwest Oncology Group (SWOG) Statistics and Data Management Center (SDMC). Assignment to S1900J is determined by the LUNGMAP protocol * Participants must have documentation of NSCLC with MET amplification determined by FMI tissue-based next generation sequencing (NGS) assay * Participants must have measurable disease documented by CT or MRI. The CT from a combined positron emission tomography (PET)/CT may be used to document measurable disease ONLY if it is of diagnostic quality: otherwise, it may be used to document non-measurable disease only. Measurable disease must be assessed within 28 days prior to sub-study registration. Pleural effusions, ascites and laboratory parameters are not acceptable as the only evidence of disease. Non-measurable disease must be assessed within 42 days prior to sub-study registration. All known sites of disease must be assessed and documented on the Baseline Tumor Assessment Form. Participants whose only measurable disease is within a previous radiation therapy port must demonstrate clearly progressive disease (in the opinion of the treating investigator) prior to sub-study registration to be considered measurable * Participants must have a CT or MRI scan of the brain to evaluate for central nervous system (CNS) disease within 42 days prior to sub-study registration * Participants with asymptomatic CNS metastasis (brain metastases or leptomeningeal disease) must be clinically stable and asymptomatic for at least 14 days prior to sub-study registration * NOTE: Participants can be on a low-dose corticosteroid treatment (≤ 10 mg prednisone or equivalent) for at least 14 days prior to study treatment * Participants must not have other known actionable oncogenic alterations, such as (but not limited to) EGFR sensitizing mutations, EGFR T790M mutation, MET Exon-14 skipping mutant NSCLC, ALK gene fusion, ROS1 gene rearrangement, RET gene rearrangement, NTRK rearrangement, HER2 mutation, KRAS activating mutations, and BRAF V600E mutation * Participants must have progressed (in the opinion of the treating physician) following the most recent line of therapy * Participants must have received at least one line of systemic treatment for Stage IV or recurrent NSCLC * Participants must have recovered (≤ Grade 1) from any side effects of prior therapy. The exception is if a side effect from a prior treatment is known to be permanent without expected further recovery or resolution (i.e., endocrinopathy from immunotherapy or cisplatin neurotoxicity) * Participants must not have been previously treated for any cancer with MET tyrosine kinase inhibitors (TKIs) such as tepotinib, capmatinib, and crizotinib * Participants must not have received any prior systemic therapy (systemic chemotherapy, immunotherapy or investigational drug) within 21 days prior to sub-study registration * Participants must not have a prior treatment with anti-PD-1 or anti-PD-L1 antibody within 6 weeks of sub-study registration * Participants must not have received any radiation therapy within 14 days prior to sub-study registration * Participants must not be planning to receive any concurrent chemotherapy, immunotherapy, biologic or hormonal therapy for cancer treatment while receiving treatment on this study * Participants must not have had major surgery excluding placement of vascular access or tumor biopsy, or had significant traumatic injury within 28 days prior to sub-study registration, or will not have fully recovered from surgery, or has surgery planned during the time the participant is expected to participate in the study * NOTE: Participants with planned surgical procedures to be conducted under local anesthesia may participate * Participants must not have a prior or concurrent malignancy whose natural history or treatment (in the opinion of the treating physician) has the potential to interfere with the safety or efficacy assessment of the investigational regimen * Absolute neutrophil count ≥ 1.5 x 10\^3/uL (within 28 days prior to sub-study registration) * Hemoglobin \>= 10.0 g/dL (within 28 days prior to sub-study registration) * Platelets ≥ 75 x 10\^3/uL (within 28 days prior to sub-study registration) * Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN) unless history of Gilbert's disease. Participants with history of Gilbert's disease must have total bilirubin ≤ 5 x institutional ULN (within 28 days prior to sub-study registration) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 3 × institutional ULN. Participants with history of liver metastasis must have AST and ALT ≤ 5 x ULN (within 28 days prior to sub-study registration) * Participants must have a serum creatinine ≤ the institutional upper limit of normal (IULN) or calculated creatinine clearance ≥ 45 mL/min using the following Cockcroft-Gault Formula. This specimen must have been drawn and processed within 28 days prior to sub-study registration. For creatinine clearance formula see the tools on the CRA Workbench * Participants' most recent Zubrod performance status must be 0-2 and be documented within 28 days prior to sub-study registration * Participants must have a completed medical history and physical exam within 28 days prior to sub-study registration * Participants with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, must have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, participants must be class 2B or better * Participants with known human immunodeficiency virus (HIV)-infection must be on effective anti-retroviral therapy and have undetectable viral load test on the most recent test results obtained within 6 months prior to sub-study registration * Participants with evidence of chronic hepatitis B virus (HBV) infection must have undetectable HBV v