Modernizing Obstetric Medicine Standards Act of 2021 or the MOMS Act This bill provides statutory authority for an existing program and establishes a new grant program within the Health Resources and Services Administration (HRSA) to improve maternal health outcomes. Specifically, the bill provides statutory authority for the Alliance for Innovation on Maternal Health program, which is administered by HRSA's Maternal and Child Health Bureau. This program supports safe maternal care by promoting the use of evidence-based practices that improve patient outcomes. HRSA must also establish a new grant program for states and hospitals to implement these practices.