Community College Student Success Act of 2021 This bill establishes two grant programs for public two-year institutions of higher education (IHEs) to plan and implement programs that are designed to increase the rate at which their students graduate, the rate at which their students transfer to a four-year IHE, and the number of credits that students apply to a degree in their major. Specifically, the bill directs the Department of Education to establish (1) a community college student success grant program, and (2) a part-time community college student success grant program. Under both programs, grant recipients may use grant funds for certain activities, including to establish career development services, expand tutoring and academic support services, and provide financial support for students. The Institute of Education Sciences must enter into a contract with an independent evaluator to evaluate these grant programs.