Black Maternal Health Momnibus Act of 2021 This bill directs multi-agency efforts to improve maternal health, particularly among racial and ethnic minority groups, veterans, and other vulnerable populations. It also addresses maternal health issues related to COVID-19 (i.e., coronavirus disease 2019). The Department of Health and Human Services (HHS) and other specified departments must address the social determinants of maternal health, which include child care, housing, food security, transportation, and environmental conditions. The bill also extends to 24 months postpartum eligibility for the Special Supplemental Nutrition Program for Woman, Infants, and Children. Additionally, HHS and other agencies must take actions to grow and diversify the maternal health workforce. To increase access to maternity care, HHS and other agencies must (1) award specified grants; (2) test an alternative payment model for perinatal care under Medicaid and the Children's Health Insurance Program (CHIP); and (3) support training, technology, and telehealth initiatives. The bill also increases research and data collection on maternal morbidity and mortality, particularly among tribal populations and minority groups. The Bureau of Prisons and the Department of Justice must establish programs to improve maternity care for incarcerated populations. The Medicaid and CHIP Payment and Access Commission must also report on the impact of Medicaid ineligibility for pregnant and postpartum prisoners. Additionally, as a condition for certain grants, states and territories must have laws limiting the use of restraints on pregnant individuals in prisons. The bill also directs activities to mitigate adverse maternal health outcomes associated with climate change and to improve maternal vaccination rates.