Puppies Assisting Wounded Servicemembers for Veterans Therapy Act or the PAWS for Veterans Therapy Act This bill implements a program and a policy related to service dog therapy for veterans. Specifically, the bill requires the Department of Veterans Affairs (VA) to implement a five-year pilot program to assess the effectiveness of addressing post-deployment mental health and post-traumatic stress disorder through a method where veterans train service dogs for veterans with disabilities. The bill also authorizes the VA to provide service dogs to veterans with mental illnesses, regardless of whether they have a mobility impairment.