Prohibiting Religious Exemption and Accommodations Databases Act This bill prohibits federal agencies from disclosing to other agencies or organizations any information relating to the religious beliefs, identity, or affiliation of an individual with a religious accommodation, including with respect to a COVID-19 vaccine requirement. The bill specifically incorporates such information into the Privacy Act of 1974 (which restricts how agencies may disclose or share records that identify individuals in the absence of written consent) and exempts related records from disclosure under the Freedom of Information Act (which provides the public a right to access federal agency information).