Black Vulture Relief Act of 2023 This bill allows livestock producers and their employees to take certain actions against black vultures to protect livestock. (Black vultures are protected under the Migratory Bird Treaty Act.) Specifically, livestock producers and their employees may (1) take (i.e., capture, kill, or disperse, or transport the carcass of) a black vulture that they reasonably believe will cause death, injury, or destruction to livestock; or (2) in the course of taking or attempting to take the black vulture, cause injury to the bird. However, they may not take or attempt to take a black vulture through the use of poison. Livestock producers and their employees must submit an annual report to the U.S. Fish and Wildlife Service (FWS) on black vultures taken under this bill. The FWS must make this reporting form available on its website. Currently, the FWS has a permit process under the Migratory Bird Treaty Act for the taking of black vultures.