Communities Serving Schools Act of 2021 This bill directs the Department of Education to award four-year competitive grants to state and local educational agencies for implementing wraparound services in public elementary and secondary schools. Wraparound service refers to a school-based service that targets academic and non-academic barriers to student achievement, including assistance with academics and physical and mental health, provision of basic needs (e.g., food, housing, and clothing), and college and career preparation. A state educational agency that receives a grant must award subgrants to local educational agencies. A local educational agency that receives a grant or subgrant must use these funds for certain activities, such as hiring an on-site coordinator to ensure wraparound services are aligned with student needs and to connect students with those services.