Combination Immunotherapy in Biochemically Recurrent Prostate Cancer

Background: Some people with prostate cancer have a rise in prostate-specific antigen (PSA). This can happen even after treatments like radiation and surgery. Androgen deprivation therapy (ADT) drugs and close monitoring are one standard way to treat this group of people. Another way is to monitor people and their PSA values over time. Researchers want to see if a combination of new drugs can help these people. Objective: To see if the combination treatment of PROSTVAC (rilimogene galvacirepvec/rilimogene glafolivec vaccinia), CV301, and MSB0011359C (M7824) can induce an anti-tumor impact in people with biochemically recurrent prostate cancer. Eligibility: People ages 18 and older with certain kinds of prostate cancer Design: Participants will be screened with * Medical history * Physical exam * Blood and urine tests * A scan of the neck, chest, abdomen, and pelvis * A bone scan A sample of tissue that was already taken will be tested. This will confirm the diagnosis, stage, and disease status. Some participants will have close monitoring with four monthly PSA checks. All participants will get two study drugs as shots under the skin. They will get the third drug in a vein. They will get the drugs over at least 7 months. Their vital signs will be checked before they get the drugs and for up to 1 hour after. Participants will have frequent study visits. They will have physical exams, urine and blood tests, and scans. Participants will return to the clinic about 4 weeks after they stop taking the study drugs. They will have a medical history, physical exam, and blood tests. They may also have long-term follow-up visits.