Ipilimumab and Nivolumab as Adjuvant Treatment of Mucosal Melanoma

Inclusion Criteria: Subject must meet all of the following applicable inclusion criteria to participate in this study: * Histological confirmation of melanoma of any mucosal site including (but not limited to) anus/rectum, vulvar/vaginal, sinonasal. NOTE: Melanomas of cutaneous origin and/or ocular origin are ineligible. * R0 or R1 resection of primary melanoma tumor (no gross disease can be left behind, but microscopically positive margins are acceptable). * Surgery within ≤ 90 days of registration. * ECOG Performance Status (PS) ≤ 1 The following laboratory values obtained ≤ 14 days prior to registration: Hematological: * Absolute Neutrophil Count (ANC) ≥ 1500/mm\^3 * Hemoglobin (Hgb) ≥ 9 g/dL (may be transfused) * Platelet (Plt) 100,000/mm\^3 Renal: * Serum Creatinine ≤ 1.5 x ULN Hepatic: * Alkaline Phosphatase (Alk Phos) ≤ 1.5 x upper limit of normal (ULN) * Total and Direct Bilirubin ≤ 1.5 × (ULN) * Aspartate aminotransferase (AST) ≤ 1.5 × ULN * Negative pregnancy test done ≤ 7 days prior to registration, for women of childbearing potential only. NOTE: Females are considered of child bearing potential unless they are surgically sterile (have undergone a hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or they are naturally postmenopausal for at least 12 consecutive months. * Written informed consent and HIPAA authorization for release of personal health information prior to registration. NOTE: HIPAA authorization may be included in the informed consent or obtained separately. * Willing to return to enrolling institution for follow-up * Willing to provide archival tissue prior to C1D1 if available and blood samples for correlative research purposes Exclusion Criteria: Subjects meeting any of the criteria below may not participate in the study: * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the subject inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens. * Immunocompromised patients and subjects known to be HIV positive and currently receiving antiretroviral therapy. NOTE: Subjects known to be HIV positive, but without clinical evidence of an immunocompromised state, are eligible for this trial. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm. * Other active malignancy ≤ 3 years prior to registration. EXCEPTIONS: Malignancies with a very low (\< 5%) risk of recurrence such as non-melanotic skin cancer or carcinoma-in-situ of the cervix. * History of myocardial infarction ≤ 6 months, or congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias. * Active autoimmune disease -including but not limited to: * Subjects with a history of inflammatory bowel disease, including ulcerative colitis and Crohn's Disease, * Subjects with a history of symptomatic autoimmune disease requiring systemic treatment within the past 2 years with the use of disease modifying agents, corticosteroids, or immunosuppressive drugs. * rheumatoid arthritis * systemic progressive sclerosis (scleroderma) * systemic lupus erythematosus * psoriasis * autoimmune vasculitis (e.g., Wegener's Granulomatosis) * CNS or motor neuropathy considered of autoimmune origin (e.g., Guillain-Barre Syndrome and Myasthenia Gravis, multiple sclerosis) EXCEPTION: autoimmune conditions that are only requiring 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. * Any radiation within 2 weeks prior to study initiation. Neoadjuvant and adjuvant radiation are allowed, but must be completed \> 2 weeks prior to registration. * Any prior systemic therapy for melanoma (chemotherapy, immunotherapy, targeted therapy) * Women of childbearing potential (WOCBP) must be willing to abstain from heterosexual intercourse or to use 2 forms of effective methods of contraception from the time of informed consent until 5 months after the last dose of study drug. The two contraception methods can be comprised of two barrier methods, or a barrier method plus a hormonal method. Examples include: intrauterine device (IUD), vasectomy of a female subject's male partner, contraceptive rod implanted into the skin, or use of two of the following: diaphragm with spermicide (cannot be used in conjunction with cervical cap/spermicide), cervical cap with spermicide (nulliparous women only), contraceptive sponge (nulliparous women only), male condom or female condom (cannot be used together), hormonal contraceptive.\*Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the subject. NOTE: Male subjects are not required to utilize contraception. The study regimen is not genotoxic and systemic concentrations sufficient to produce a risk of fetal toxicity are not expected in WOCBP partners from exposure to a male participant's seminal fluid. * Pregnant or breastfeeding (NOTE: breast milk cannot be stored for future use while the mother is being treated on study).