VA Suicide Prevention Services Accountability Act This bill requires the Government Accountability Office (GAO) to conduct a management review and submit reports on mental health and suicide prevention efforts carried out by the Department of Veterans Affairs (VA). Specifically, the bill requires the GAO to conduct a management review of the mental health and suicide prevention services provided by the VA. The bill also requires the GAO to report on the VA's efforts to (1) manage veterans at high risk for suicide, (2) integrate mental health care into VA primary care clinics, and (3) integrate community-based mental health care (care provided by a non-VA provider but paid for by the VA) into the Veterans Health Administration.