Rural Veterans Travel Enhancement Act of 2021 This bill addresses Department of Veterans Affairs (VA) transportation benefits for certain veterans. First, the bill makes permanent the authority for the VA to transport any person to or from a VA facility or other place in connection with vocational rehabilitation, required counseling, or for an examination, treatment, or care. The bill also permanently extends the authorization of appropriations for grants provided to state veterans service agencies and veterans service organizations to transport highly rural veterans for medical care. The bill requires the Government Accountability Office (GAO) to study fraud, waste, and abuse of benefits under the VA beneficiary travel program and report the findings to Congress. The GAO must also study and report on (1) the efficacy of the current mileage reimbursement rate under the program in mitigating the financial burden of transportation costs for traveling to and from VA medical facilities for care, (2) the origins of the amount of the deductible and its impact on the efficacy of the benefits provided in mitigating the financial burden on veterans seeking medical care, and (3) recommendations for how this program or another program could further encourage veterans to seek medical care. The bill also requires the VA to implement various pilot programs related to transportation benefits for veterans for specified purposes.