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 stan…