Veterans Pro Bono Corps Act of 2021 This bill requires the Department of Veterans Affairs (VA) to implement a five-year pilot program to assess the feasibility and advisability of providing assistance to eligible recipients to establish or maintain programs that assist veterans applying for compensation for a service-connected disability. Eligible recipients are accredited entities that are administering a medical residency or medical fellowship program in the United States or a territory of the United States. Grants disbursed under the pilot program must be used to establish or maintain programs in which medical residents or fellows provide pro bono medical examinations and opinions to assist veterans in applying for service-connected disability benefits. The VA must partner with veterans service organizations to provide outreach regarding the availability of services under the pilot program.