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 provide canine training to eligible veterans diagnosed with post-traumatic stress disorder as an element of a complementary and integrative health program. Eligible veterans are those who are enrolled in the VA health care system and have been recommended for participation by a qualified mental health care provider or clinical team. The VA must seek to enter agreements containing specified elements with accredited nongovernmental entities that have demonstrated ability to provide canine training. Veterans who participate in the program are authorized to adopt the dog they assisted in training if their health provider determines it is in the best interest of the veteran. The VA must establish processes to document and track the progress of participating veterans with respect to health benefits and improvements. The Government Accountability Office must brief Congress and submit a report on the program. The bill also authorizes the VA to provide service dogs to veterans with mental illnesses, regardless of whether they have a mobility impairment.