The purpose of this study is to test the effectiveness, safety, and tolerability of the drugs nivolumab plus ipilimumab with or without the addition of stereotactic body radiation therapy (SBRT). Nivolumab is an antibody (a type of human protein) that is being tested to see if it will stimulate the body's immune system to work against tumor cells. This study will test an investigational use of nivolumab.

Inclusion Criteria:

• At least 18 years of age
• Eastern Cooperative Oncology Group (ECOG) Performance Status less than 2
• Active disease measurable by CT, MRI or clinical exam.
• Prior chemotherapy or immunotherapy will be allowed if new or persistent measurable site(s) of disease are present.
• Prior radiation therapy will be allowed if there is active measurable disease burden.
• Must be either recurrent, unresectable or Stage IV American Joint Committee on Cancer (AJCC) (7th edition) and have histologically confirmed Merkel cell carcinoma with at least 2 distinct lesions in order to be eligible.
• Must have at least 2 distinct lesions as documented by a complete physical examination or imaging studies within 4 weeks prior to randomization. Imaging studies must include a diagnostic CT scan of the involved disease sites and all known sites of resected disease and brain magnetic resonance (MRI) or CT (brain CT allowable if MRI is contraindicated or if there is no known history of resected brain lesions).
• Tumor tissue from the core biopsy or resected site of disease must be provided for biomarker analyses.

Exclusion Criteria:

• History of Grade 3 toxicity or use of infliximab with prior immunotherapy
• Patients with active brain metastasis.
• Active, known, or suspected autoimmune disease. Potential participants with type I diabetes mellitus, residual hypothyroidism due to autoimmune thyroiditis only requiring hormone replacement, skin disorders (such as vitiligo, psoriasis, or alopecia) not requiring systemic treatment are permitted to enroll.
• Patients with prior history of non-Merkel cell carcinoma malignancies are excluded except adequately treated basal cell, squamous cell skin cancer, chronic lymphocytic leukemia or other indolent diseases not requiring therapy; adequately treated, with curative intent, cancer from which the patient is currently in complete remission per investigator's judgment; or patients with history of breast cancer and no evidence of disease on hormonal therapy to prevent recurrence and patients with prostate cancer on adjuvant hormonal therapy with undetectable PSA are eligible.
• A condition requiring systemic treatment with either corticosteroids (> 10 mg daily prednisone equivalent) or other immunosuppressive medications within 14 days of randomization. Inhaled or topical steroids are permitted in the absence of active autoimmune disease.

Adverse events collected from on study date to 100 days after end of treatment (an average of 10.5 months). Survival will be assessed up to 5 years.