Housing Fairness Act of 2021 This bill reauthorizes through FY2032 the Fair Housing Initiatives Program and revises program reporting requirements and organization eligibility requirements. The bill also allows the Department of Housing and Urban Development (HUD) to enter into agreements with certain organizations to test and investigate differential treatment and discriminatory housing practices based on race, color, sex, religion, national origin, disability, and familial status. The results of these tests and investigations may be used as the basis for an enforcement action under specified fair housing laws. Additionally, HUD must establish a grant program for public and private nonprofit organizations to study housing discrimination and implement pilot projects to test solutions.