Accessible Voting Act of 2021 This bill addresses voting accessibility for individuals with disabilities and older individuals. First, the bill establishes the Office of Accessibility within the Election Assistance Commission (EAC). The office must serve as the advocate on behalf of voters with access needs, ensure state and local election officials are maintaining voting systems that meet Department of Justice accessibility standards, and make payments to states to implement the accessibility requirements outlined by the bill. In addition, the office must establish and operate a National Resource Center on Accessible Voting to provide state and local election officials, poll workers, and volunteers with the information and technical assistance needed to provide accessible voting. The office must also work with state and local election officials to collect data and administer a public national voter accessibility website. The bill requires each state to (1) ensure election websites are accessible and meet specified requirements, (2) designate a single office to provide information on federal elections to individuals with disabilities and older individuals, (3) notify the public of changes to voting practices, (4) expand access to absentee registration procedures and no-excuse mail-in absentee voting in federal elections, and (5) ensure all polling places and voter drop boxes are accessible. The EAC must coordinate with specified agencies to make grants to eligible entities to study, test, and develop accessible voting options. The bill also permanently reauthorizes and expands a Department of Health and Human Services grant program for increasing accessibility to polling places.