Fairness for Local Veteran Cemeteries Act This bill authorizes the Department of Veterans Affairs (VA) to provide a grant under the Veteran Cemetery Grants Program to a county to establish, expand, improve, operate, or maintain a county-owned veterans' cemetery. Under current law, such grants are only authorized to be issued to states, tribal organizations, and U.S. territories or possessions. The VA is only able to provide such grants under certain circumstances. Specifically, grants are only authorized if the state in which the county is located does not have a state-owned veterans' cemetery or the state did not apply for a grant in the previous year. Additionally, a county must demonstrate that it has the resources necessary to operate and maintain its cemetery. The VA must give priority to state-owned cemeteries in situations where both the county and the state apply for a grant in the same year.