Strengthening the Vaccines for Children Program Act of 2021 This bill modifies vaccination programs and coverage for children under Medicaid and the Children's Health Insurance Program (CHIP). Specifically, the bill expands and otherwise modifies the Vaccines for Children Program, which provides free vaccinations to individuals 18 years of age or younger who are uninsured or underinsured (i.e., with limited or no insurance for vaccines), are eligible for Medicaid, or are American Indians or Alaskan Natives. The bill expands the program to include children enrolled in CHIP and temporarily provides for additional payments to incentivize provider participation. The Government Accountability Office must report on the related effects of the bill's changes. The bill also expands coverage under Medicaid and CHIP to include vaccine counseling and educational services for children and temporarily requires payments to be made at a rate that is no less than the rate under Medicare for equivalent services. It also temporarily increases the Federal Medical Assistance Percentage (i.e., federal matching rate) for states that conduct culturally appropriate outreach regarding the benefits of vaccinations for children.