National Instant Criminal Background Check System (NICS) Data Integrity Act of 2021 This bill modifies procedures for conducting firearms-related background checks through the national instant criminal background check system (NICS). Specifically, the bill requires the NICS to retain records related to a firearms-related background check for as long as necessary to determine whether an individual is eligible to receive a firearm. Additionally, the bill requires the NICS to query additional databases for relevant information, including the National Data Exchange, the Interstate Identification Index System, and the National Crime Information Center databases.