This phase II trial tests whether nivolumab in combination with cabozantinib works in patients with mucosal melanoma. Immunotherapy with monoclonal antibodies, such as nivolumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Cabozantinib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. It works by blocking the action of an abnormal protein that signals tumor cells to multiply. This helps stop the spread of tumor cells.