Increasing Access to Quality Cardiac Rehabilitation Care Act of 2021 This bill accelerates and expands certain changes that are scheduled to take effect relating to the coverage of cardiac, intensive cardiac, and pulmonary rehabilitation programs under Medicare. Currently, such programs require physician supervision. Effective January 1, 2024, physician assistants, nurse practitioners, and clinical nurse specialists may also supervise these programs. The bill expedites the effective date of these changes to January 1, 2022, and also allows these practitioners to administer programs in their offices, prepare and sign treatment plans, and prescribe exercise.