Inclusion Criteria

Histologically confirmed unresectable or metastatic translocation morphology renal cell carcinoma diagnosed using World Health Organization (WHO)-defined criteria. Patients may be newly diagnosed or have received prior cancer therapy

Patients must have had histologic verification of the malignancy

Patients must have measurable disease, documented by clinical, radiographic, or histologic criteria

Patients must have a tumor showing the appropriate morphologic appearance, and either confirmed TFE3 nuclear protein expression by immunohistochemistry with appropriate positive and negative controls performed at a Clinical Laboratory Improvement Act (CLIA)-certified laboratory, or evidence of TFE3 or TFEb translocation by either fluorescence in situ hybridization (FISH) or reverse transcriptase- polymerase chain reaction (RT-PCR) performed at a CLIA-certified laboratory. For TFE3 immunohistochemistry, any nuclear positivity in the presence of appropriate positive and negative controls should be considered as evidence of TFE3 immunohistochemical expression. NOTE: If the institution is unable to perform these studies, unstained slides may be submitted to Dr. Elizabeth Perlman, who will perform TFE3 analysis at no charge. The slide will be returned to the referring institution for local evaluation, to be included in their institutional report

Patients must have a life expectancy of >= 8 weeks

Patients must have fully recovered from the acute toxic effects of all prior chemotherapy, immunotherapy, or radiotherapy prior to entering this study

Myelosuppressive chemotherapy: Must not have received within 2 weeks of entry onto this study (6 weeks if prior nitrosourea)

Immunotherapy: Must not have received within 4 weeks of entry onto this study

Biologic (anti-neoplastic agent): At least 7 days since the completion of therapy with a biologic agent

Radiation therapy (RT): >= 2 weeks for local palliative RT (small port); >= 6 months must have elapsed if prior craniospinal RT or if >= 50% radiation of pelvis; >= 6 weeks must have elapsed if other substantial bone marrow (BM) radiation

Prior therapy with tyrosine kinase inhibitors (TKIs) that have no known activity against VEGFR1-3 are permitted

No history of myocardial infarction, severe or unstable angina, or peripheral vascular disease

Exclusion Criteria

Patients unable to swallow whole tablets

Patients who in the opinion of the investigator are not able to comply with the study procedures are not eligible

Patients who have received prior therapy with axitinib or nivolumab or other VEGF or PD1/PD-L1 targeted therapies

Patients with hypersensitivity to axitinib, nivolumab, or any of its excipients

Patients who previously received an allogeneic stem cell transplant (SCT) or solid organ transplant are not eligible

Patients may not be receiving any other investigational agents (within 4 weeks prior to study enrollment)

Patients who have received prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study enrollment or who have not recovered (i.e., =< grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks prior to enrollment

Patients who have had or who are planning to have the following invasive procedures are not eligible:

- Major surgical procedure, laparoscopic procedure, open biopsy, core biopsy, fine needle aspirate, or significant traumatic injury within 7 days prior to enrollment. NOTE: External central lines must be placed at least 3 days prior to planned treatment initiation and subcutaneous ports must be placed at least 7 days prior to planned treatment initiation

Patients who have a serious or non-healing wound or ulcer at the time of study enrollment are not eligible

Patients who have a history of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 28 days of study enrollment are not eligible

Patients who have received prior targeted small molecule therapy within 2 weeks of enrollment or have not recovered (i.e., =< grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks prior to enrollment. NOTE: Subjects with =< grade 2 neuropathy are an exception to this criterion and may qualify for the study

Pre-existing conditions, which may include:

- Additional known malignancy that is progressing or requires active treatment. Exceptions include basal cell carcinoma of the skin, or squamous cell carcinoma of the skin that has undergone potentially curative therapy, or in situ cervical cancer

- Patients with underlying immune deficiency, chronic infections including hepatitis, tuberculosis (TB), or autoimmune disease

- Human immunodeficiency virus (HIV)-infected patients with the exception of patients on an effective anti-retroviral therapy with an undetectable viral load within 6 months prior to enrollment

- Patients with underlying hematologic issues including congenital bleeding diathesis, known previous gastrointestinal (GI) bleeding requiring intervention within the past 6 months, history of hemoptysis within 42 days prior to study enrollment, active pulmonary emboli, or deep vein thromboses (DVT) that are not stable on anticoagulation regimen

Patients must not have had significant vascular disease (i.e. Moya-Moya, aortic aneurysm requiring surgical repair)

A known history of, or any evidence of active, non-infectious pneumonitis

Patients with known active central nervous system (CNS) metastases and/or carcinomatous meningitis or leptomeningeal disease. Patients with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least 4 weeks prior to study enrollment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to study enrollment. This exception does not include carcinomatous meningitis which is excluded regardless of clinical stability

Any uncontrolled, intercurrent illness including but not limited to ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia

Any serious medical or psychiatric illness/condition including substance use disorders likely in the judgment of the investigator(s) to interfere or limit compliance with study requirements/treatment

Patients with active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment

Treatments and/or medications the patient is receiving or has received that would make her/him ineligible, including:

- Concomitant (or receipt of) treatment with medications that may affect the metabolism of nivolumab and/or axitinib within 7 days prior to planned first dose of protocol therapy

- A live vaccine within 30 days of planned first dose of protocol therapy. NOTE: Inactivated flu vaccines are allowed; however intranasal influenza vaccines (e.g., Flu-Mist) are live attenuated vaccines, and are not allowed

Due to risks of fetal and teratogenic adverse events as seen in animal studies, a negative pregnancy test must be obtained in females of childbearing potential, defined as females who are post-menarchal. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required

Females of childbearing potential that are sexually active must agree to either practice 2 medically accepted highly-effective methods of contraception at the same time or abstain from heterosexual intercourse from the time of signing the informed consent through 5 months after the last dose of study drug

Lactating females are not eligible unless they have agreed not to breastfeed their infants starting with the first dose of study therapy through 5 months after the last dose of study therapy

Male patients must agree to use an adequate method of contraception starting with the first dose of study therapy through 7 months after the last dose of study therapy. Prior history of vasectomy does not replace requirement for contraceptive use