Improve Well-Being for Veterans Act This bill requires the Department of Veterans Affairs (VA) to award grants for a period of three years to eligible entities for the provision of suicide prevention services to veterans and their families. Under the bill, eligible entities include tribally designated housing entities, community-based organizations, and other private entities that meet certain requirements. Entities must submit an application to receive financial assistance that includes a description of the services they propose to provide, among other elements. The bill requires the VA to equitably distribute grants across geographic regions and areas with the most need. Additionally, the VA shall give preference to entities that are experienced in providing suicide prevention services. Entities receiving funding must notify service recipients that funding is wholly or partially from the VA.