Family Reunification Act of 2021 This bill addresses issues related to aliens obtaining legal permanent resident status, including by removing certain deadlines in a provision that allows eligible aliens to obtain such status. Specifically, the bill amends a provision that allows an alien to obtain lawful permanent resident status if the alien (1) is physically present in the United States, (2) entered the United States without inspection or meets certain other criteria, (3) is a beneficiary of a qualifying family-based immigrant petition or a labor certification application filed no later than April 30, 2001, and (4) has filed an application and meets other requirements. This bill removes the April 30, 2001, deadline for the filing of the required immigrant petition or labor certification. It also removes another requirement for certain applying aliens to have been physically present in the United States on December 21, 2000. Furthermore, a beneficiary of a family-based immigrant petition that appears to be eligible for approval may not be removed while the petition is pending. The bill also makes nonimmigrant V visas available to a beneficiary of an approved family-based immigrant petition, subject to certain requirements, regardless of when the immigrant petition was filed or how long it has been pending. Currently, a V visa is only available to an alien whose approved family-based immigrant petition was filed no later than December 11, 2000, and has been pending for at least three years. A V visa holder may not receive any means-tested public benefits or certain health insurance-related benefits.