Senior Access to Vaccines Ensured Act of 2021 or the SAVE Act of 2021 This bill requires the Centers for Medicare & Medicaid Services (CMS) to establish a program that provides nonemergency medical transportation (NEMT) for Medicare beneficiaries to COVID-19 vaccination sites during the public health emergency. The CMS must contract with qualified transportation managers to implement the program, with preference given to those with prior experience under a state Medicaid NEMT program. The CMS must also establish criteria for beneficiary participation that is similar to the criteria used by states for Medicaid NEMT services.