Nivolumab and Radiation Therapy or Ipilimumab as Adjuvant Therapy in Treating Patients With Merkel Cell Cancer

Inclusion Criteria: * Be willing and able to understand and give written informed consent and comply with all study related procedures * All patients should undergo definitive surgical resection, including when possible sentinel lymph node dissection * Patients must have recovered after any recent surgery and be ambulatory * Have node positive disease (stage pIIIA or pIIIB) +/- extracapsular extension * Have node negative disease and any of the following high risk features * Tumor size \>= 2 cm * Margins =\< 1-2 cm and re-resection is not possible * Evidence of perineural or lymphovascular invasion * Human immunodeficiency virus (HIV) patients with undetectable viral load and CD4+ T-cell counts \>= 350 cells/uL * Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 * Absolute neutrophil count (ANC) \>= 1,500/mcL (performed within 16 days of treatment initiation) * Platelets \>= 100,000/mcL in the absence of transfusion support within 7 days of determining eligibility (performed within 16 days of treatment initiation) * Hemoglobin \>= 8 g/dL (performed within 16 days of treatment initiation) * Serum creatinine =\< 1.5 x upper limit of normal (ULN) OR Measured or calculated creatinine clearance \>= 40 mL/min creatinine clearance (performed within 16 days of treatment initiation) (Glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinine clearance \[CrCl\]) * Creatinine clearance should be calculated per institutional standard * Serum total bilirubin =\< 1.5 x ULN OR Except subjects with Gilbert Syndrome, who can have total bilirubin \< 3.0 x ULN (performed within 16 days of treatment initiation) * Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT)(serum glutamate pyruvate transaminase \[SGPT\]) =\< 3.0 x ULN (performed within 16 days of treatment initiation) * Female subject of childbearing potential should have a negative urine or serum pregnancy at screening and within 24 hours prior to receiving the first dose of study medication and then every 4 weeks while on treatment. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Female subjects of childbearing potential should be willing to use 2 methods of birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study through 5 months after the last dose of study medication. Subjects should agree to ongoing pregnancy testing during the course of the study and after the end of study therapy. Female subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year * Male subjects should 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. Males must refrain from donating sperm during study participation and for 7 months after the last dose of study medication. Female subject should agree to use an adequate method of contraception starting with the first dose of study therapy through 5 months after the last dose of study therapy Exclusion Criteria: * Has known distant metastatic MCC * Has a known history of active TB (Bacillus tuberculosis) * Hypersensitivity to any study agents * Have received prior immunotherapy with any PD-1/PDL-1 inhibitors or CTLA-4 antibodies at any time in the past * Has had prior chemotherapy or radiation therapy for treatment of MCC * Has a clinically significant medical condition, which in the judgment of the attending physician would contraindicate immunotherapy or radiotherapy, such as serious autoimmune disease, hypersensitivity to investigational product or any component in its formulations, per Food and Drug Administration (FDA) prescription notice * Subjects with prior history of non-Merkel cell carcinoma malignancies are excluded except * Adequately treated basal cell, squamous cell skin cancer, chronic lymphocytic leukemia (CLL) or other indolent malignancies not requiring therapy (ie. active surveillance) * Adequately treated malignancies and patient has been in complete remission for at least two years * Patients with history of breast cancer and no evidence of disease on hormonal therapy to prevent recurrence * Patients with prostate cancer on adjuvant hormonal therapy with undetectable PSA are eligible * Has 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 (thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment * Subjects with a condition requiring systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalent) or other immunosuppressive medications within 14 days of randomization will be excluded. Inhaled or topical steroids are permitted in the absence of active autoimmune disease * Has an active infection requiring intravenous systemic therapy * Solid organ transplant recipients and patients with concurrent hematological malignancies including thymomas, leukemias (other than CLL) and lymphomas actively undergoing treatment or completed \< 5 years prior * Clinically significant cardiovascular disease with uncontrolled arrhythmia, New York Association class 3 or 4 congestive heart failure, history of myocardial infarction within 6 months, or prolonged corrected QT (QTc) \> 500 msec * Is pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment * Has known active hepatitis B (e.g., hepatitis B virus surface antigen \[HBsAg\] reactive) or hepatitis C (e.g., hepatitis C virus \[HCV\] ribonucleic acid \[RNA\] \[qualitative\] is detected)