This phase II trial tests how well nivolumab and ipilimumab immunotherapy with or without cabozantinib works in treating patients with nasopharyngeal cancer that has come back (after a period of improvement) (recurrent), has spread from where it first started (primary site) to other places in the body (metastatic), or for which no treatment is currently available (incurable). Immunotherapy with monoclonal antibodies, such as nivolumab and ipilimumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Cabozantinib is in a class of medications called kinase inhibitors. It works by blocking the action of an abnormal protein that signals cancer cells to multiply. This helps slow or stop the spread of cancer cells. Giving immunotherapy with nivolumab and ipilimumab and targeted therapy with cabozantinib may help shrink and stabilize nasopharyngeal cancer.

PRIMARY OBJECTIVE: I. To determine if the progression-free survival (PFS) of the triplet combination (cabozantinib S-malate, nivolumab, and ipilimumab \[CaboNivoIpi\]) is more favorable than the doublet (nivolumab and ipilimumab \[NivoIpi\]). SECONDARY OBJECTIVES: I. To compare safety and tolerability between the two arms (Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v\]5.0.). II. To compare overall response rate (ORR) between the two arms via both Response Evaluation Criteria in Solid Tumors (RECIST) v1.1 and immune-modified Response Evaluation Criteria in Solid Tumors (iRECIST) criteria. III. To compare overall survival (OS) between the two arms. IV. To assess response by primary or acquired PD-1/L1 inhibitor resistance in the prior line of therapy. EXPLORATORY OBJECTIVE: I. To evaluate molecular and immunologic predictors of response (Epstein-Barr virus \[EBV\] viral load; PD-L1 score) between arms. OUTLINE: Patients are randomized to 1 of 2 arms. ARM A: Patients receive nivolumab intravenously (IV) over 30 minutes and ipilimumab IV over 30 minutes on day 1 of each cycle. Cycles repeat every 28 days for up to 2 years in the absence of disease progression or unacceptable toxicity. Patients undergo computed tomography (CT) or magnetic resonance imaging (MRI) and collection of blood samples throughout the trial. ARM B: Patients receive nivolumab IV over 30 minutes and ipilimumab IV over 30 minutes on day 1 and cabozantinib S-malate orally (PO) daily on days 1-28 of each cycle. Cycles repeat every 28 days for up to 2 years in the absence of disease progression or unacceptable toxicity. Patients may continue with cabozantinib S-malate after 2 years per treating investigator. Patients undergo CT or MRI and collection of blood samples throughout the trial. After completion of study treatment, patients are followed up every 8-12 weeks until progression of disease occurs or a new non-protocol anti-cancer therapy is initiated and then every 6 months for up to 2 years.