Conviction Integrity Act of 2021 This bill requires the Department of Justice to administer grant programs within the Bureau of Justice Assistance (BJA) to encourage the review of possible cases of wrongful conviction and facilitate post-conviction relief. Specifically, the bill directs the BJA to make grants to create state and local conviction integrity units or entities that receive petitions for or review wrongful convictions and wrongful sentences. Additionally, the BJA must make grants to support high-quality post-conviction legal representation for defendants who make post-conviction claims of innocence.