No Vaccine Passports Act This bill prohibits certain actions related to vaccine passports and proof of COVID-19 vaccination. Specifically, a federal agency may not issue a vaccine passport, vaccine pass, or other standardized documentation for the purpose of certifying the COVID-19 vaccination status of a U.S. citizen to a third party, or otherwise publish or share any COVID-19 vaccination record of a U.S. citizen or similar health information. Additionally, proof of COVID-19 vaccination shall not be required to access federal or congressional property or services.