Vaccine Access Improvement Act of 2021 This bill modifies the excise tax on certain vaccines to (1) automatically impose the tax on vaccines that the Department of Health and Human Services (HHS) adds to the Vaccine Injury Table, and (2) require HHS to notify the Department of the Treasury and Congress when new vaccines are added to the table. (The excise tax funds the National Vaccine Injury Program, which compensates people who have been injured by vaccines listed on the table.)