No Vaccine Mandates Act of 2023 This bill creates a federal statutory framework to govern restrictions and requirements related to COVID-19 vaccinations. First, the bill temporarily prohibits (1) requiring an individual to receive a COVID-19 vaccine, or (2) administering a COVID-19 vaccine to a minor or other individual who lacks the capacity to consent without the consent of a parent or guardian. A violation is subject to criminal penalties. Second, the bill requires any person who administers a COVID-19 vaccine to provide a potential recipient with certain information about the risks associated with the vaccine so the potential recipient can make an informed decision. Third, the bill generally prohibits the public disclosure of an individual's COVID-19 vaccination status without express, written consent. A violation is subject to criminal and civil penalties. This bill does not preempt state or local laws that provide greater privacy protection than these provisions.