Affordable Housing for Homeless Veterans Act of 2021 This bill reauthorizes and modifies a program of the Department of Veterans Affairs (VA) that provides housing assistance to homeless veterans. Specifically, the bill authorizes the VA to enter into agreements with specified entities (e.g., nonprofit organizations or tribal entities) through September 30, 2025, to assist homeless veterans and their families in acquiring shelter or permanent housing. Under the bill, the VA may enter into agreements where an entity that is buying, leasing, or receiving property agrees to sell or rent the property directly to homeless veterans or veterans at risk of homelessness.