Veterans Pro Bono Corps Act of 2021 This bill requires the Department of Veterans Affairs (VA) to study the feasibility and advisability of establishing a five-year pilot program to provide assistance to entities that provide aid to veterans applying for compensation for a service-connected disability, specifically by helping them substantiate their claims with independent medical examinations and opinions. The study must also assess the use of the VA disability benefits questionnaire forms to support the pilot program. Under the program, eligible entities are those that administer a non-VA medical residency or medical fellowship program in the United States (or a territory of the United States) and are accredited by a nationally recognized accrediting agency.