This phase I/II trial studies the side effects of infliximab and vedolizumab and to see how well they work in treating inflammation of the colon (colitis) caused by immune checkpoint inhibitor therapy in patients with cancer of the genital and urinary organs (genitourinary) or melanoma. Monoclonal antibodies, such as infliximab or vedolizumab, may help to treat immunotherapy induced colitis/diarrhea. This study may help to identify the optimal treatment strategy for immune checkpoint inhibitor-related colitis in patients with genitourinary cancer or melanoma.