No Vax Pass Act of 2021 This bill generally prohibits states and localities from expending COVID-19 relief funds on systems (other than in health care contexts) that verify an individual's COVID-19 vaccination status. However, the bill provides an exception for state educational agencies, local educational agencies, institutions of higher education, secondary schools, and elementary schools. These entities may implement a system to verify the COVID-19 vaccination status of students and staff provided that requirements for COVID-19 vaccines are no stricter than requirements for any other vaccine.