Full-Service Community School Expansion Act of 2021 This bill reauthorizes through FY2025 and expands the Full-Service Community Schools program. The program provides support for full-service community schools that serve children and their families, particularly for children who attend high-poverty elementary and secondary schools. Specifically, the bill directs the Department of Education (ED) to award grants to eligible entities to plan, implement, expand, and support full-service community schools. An eligible entity is a consortium of (1) one or more local educational agencies or the Bureau of Indian Education; and (2) one or more community-based organizations, nonprofit organizations, Indian tribes, tribal organizations, or other entities. A community school refers to a public elementary or secondary school that incorporates the four pillars of community schools. These four pillars are defined by the bill as (1) integrated student supports that address out-of-school barriers to learning (e.g., medical care and assistance with housing, transportation, or nutrition); (2) expanded and enriched learning time and opportunities (e.g., before- and after-school programs); (3) family and community engagement (e.g., adult education); and (4) collaborative leadership and practices, which must include a school-based leadership team, a community school coordinator, and a community-wide leadership team. Further, the bill directs ED to award grants to states to plan, implement, and expand community schools. A state educational agency that receives a grant must meet certain requirements, such as providing subgrants to local educational agencies, establishing a state-level steering committee, and establishing goals for the implementation and expansion of community schools throughout the state.