Debt-Free College Act of 2021 This bill establishes measures to cover the unmet financial need of students who are enrolled at certain institutions of higher education (IHEs). Unmet financial need refers to the difference between a student's cost of attendance and the student's expected family contribution, plus any federal, state, and local sources of grant aid. In addition, the bill makes certain Dreamer students (i.e., students who have been granted Deferred Action for Childhood Arrivals status) eligible for federal financial aid. First, the bill requires the Department of Education (ED) to award grants for state-federal partnerships with a goal of providing debt-free college for all eligible students at in-state public IHEs. Eligible student refers to an individual who (1) is enrolled or is eligible to enroll in an in-state public IHE, (2) completes a Free Application for Federal Student Aid or demonstrates eligibility for a Federal Pell Grant through institutional financial-aid eligibility forms, and (3) demonstrates satisfactory academic progress. Next, the bill requires ED to award grants to historically Black colleges and universities and minority-serving institutions to cover the unmet financial need of enrolled students. ED must establish an office to administer grants and provide oversight. In addition, the bill makes Dreamer students who entered the United States before the age of 16 and who meet certain educational criteria eligible for federal student aid.