Huntington's Disease Disability Insurance Access Act of 2021 This bill expedites payment of Social Security Disability Insurance (SSDI) benefits and eligibility for Medicare coverage for those with Huntington's disease. This is an inherited disorder that causes brain cells to die, including in areas of the brain that help to control voluntary movement. Specifically, the bill eliminates the 5-month waiting period for SSDI benefits and the subsequent 24-month waiting period for Medicare coverage for individuals with Huntington's disease. Under current law, individuals generally must wait five months after the onset of disability to begin receiving SSDI benefits and an additional 24 months to become eligible for Medicare.