Election Security Act of 2022 This bill addresses election security through grant programs and requirements for voting systems and paper ballots. Among other provisions, the bill (1) establishes requirements for voting systems, including that systems use individual, durable, voter-verified paper ballots; (2) directs the Election Assistance Commission to award grants to states for specified activities, including replacing voting systems and improving the security of the systems; and (3) requires states and jurisdictions to carry out postelection audits for all federal elections.