Emergency Grants of Release And Compassion Effectively Act of 2021 or the Emergency GRACE Act This bill sets forth provisions to prevent and limit the spread of COVID-19 in federal correctional facilities (e.g., prisons) and state prison systems. Among the provisions, the bill expands access to temporary release on furlough and compassionate release for federal prisoners during a public health emergency; allows Medicaid payment for medical services furnished to an incarcerated individual during the 30-day period preceding the individual's release; and requires federal correctional facilities, including contract facilities, to follow the procedures recommended by the Centers for Disease Control and Prevention for limiting the spread of the coronavirus. Additionally, the bill provides FY2021 and FY2022 supplemental appropriations to help state prison systems expand testing of inmates and facilitate the compassionate release of high-risk inmates. The supplemental appropriations are designated as emergency spending, which is exempt from discretionary spending limits.