Veterans Entrepreneurship Act of 2021 This bill requires the Small Business Administration to carry out a pilot program assessing the feasibility and advisability of making grants to eligible veterans to start or acquire a small business concern, a franchise business enterprise, or any other appropriate category of business enterprise. Eligible veterans are those veterans or members of the Armed Forces who (1) are currently retiring or separating from the Armed Forces or have been honorably discharged from active duty, (2) have completed a specified minimum period of active duty, and (3) are entitled to certain educational assistance programs. Grant recipients must complete an entrepreneurship readiness program and prepare a business plan, and they may use grant funds for purposes such as purchasing goods or services, funding a project that is directed toward economic development objectives, and acquiring a business enterprise.