Veteran Wellness Act This bill requires the Department of Veterans Affairs (VA) to carry out a two-year pilot program under which the VA makes grants to nonprofit veterans services organizations (VSOs), such as the American National Red Cross, the American Legion, the Disabled American Veterans, the United Spanish War Veterans, and the Veterans of Foreign Wars, to: (1) upgrade their community facilities into health and wellness centers in at least 10 different geographic locations, and (2) promote and expand complementary and integrative wellness programs. In selecting grant recipients, the VA shall give priority to recipients that will use grant amounts in locations in economically depressed areas that are not in close proximity to VA medical centers. A VSO, as a condition of receiving a grant, must agree to carry out the operation and maintenance of a facility that is repaired or constructed using grant funds for three years after completion of the repair or construction.