Women Veterans Support Act This bill implements programs and studies to improve mental health care for veterans, with a focus on women veterans. Specifically, the Department of Veterans Affairs (VA) must enter into an agreement with an independent entity to conduct a study on suicide by women veterans. The bill requires the VA to perform outreach on a national, regional, and local scale to prevent suicide by women veterans. The VA must establish a five-year grant program for certain nonprofit organizations to provide specified services to women veterans. Such services include, but are not limited to, mental health support services and financial counseling services. The bill requires the VA to provide financial assistance to eligible entities to provide or coordinate the provision of specified mental health supportive services for a veteran with a mental health condition who is seeking mental health treatment. Such entities must apply, and the VA shall provide training and technical assistance to entities regarding the planning, development, and provision of such services. For purposes of this financial assistance program, eligible entities include tribally designated housing entities and certain private entities (e.g., incorporated private institutions). The bill authorizes the VA to collect information from a participating entity relating to a mental health condition of a veteran for inclusion in the electronic health record of the VA in order to improve care provided to such veteran. The bill requires the VA to implement, with a specified framework, the Department of Veterans Affairs Readjustment Counseling Service Scholarship Program.