Veterans, Women, Families with Children, Race, and Persons with Disabilities Housing Fairness Act of 2021 or the Housing Fairness Act of 2021 This bill expands efforts to detect and address housing discrimination. Specifically, the Department of Housing and Urban Development (HUD) must conduct a nationwide testing program to (1) detect and document differences in the treatment of prospective renters, homebuyers, or mortgage borrowers; (2) measure patterns of adverse treatment because of the race, color, religion, sex, familial status, disability status, or national origin of a renter, homebuyer, or borrower; and (3) measure the prevalence of such discriminatory practices across housing and mortgage lending markets. The bill also reauthorizes through FY2025 the Fair Housing Initiatives Program, which supports organizations that provide direct assistance to individuals who have been victims of housing discrimination. Additionally, HUD shall implement a grant program to assist public and private nonprofit organizations in (1) conducting comprehensive studies of the causes and effects of housing discrimination and segregation on education, poverty, and economic development or on veterans and military personnel; and (2) implementing pilot projects that test solutions to help prevent or alleviate housing discrimination and segregation.