Accidental Firearms Transfers Reporting Act of 2021 This bill requires the Federal Bureau of Investigation (FBI) to report to Congress on the number of firearm transfers where (1) a licensed gun dealer submits a background check through the National Instant Criminal Background Check System (NICS), (2) NICS fails to respond within three business days, and (3) the licensed gun dealer completes the transfer. The FBI must also report on the procedures followed once the transferee is discovered to be ineligible.