Veterans Dog Training Therapy Act This bill directs the Department of Veterans Affairs (VA) to carry out a five-year pilot program to assess the effectiveness of addressing veterans' post-deployment mental health and post-traumatic stress disorder symptoms through the therapeutic medium of training service dogs for veterans with disabilities. The VA shall: seek to enter into contracts with non-government entities that are certified in the training and handling of service dogs and that are located in close proximity to at least three but not more than five VA medical centers, provide such entities access to a training area in a VA facility that would be appropriate for use in educating veterans with mental health conditions in service dog training and handling, administer the program through the Center for Compassionate Innovation of the Veterans Health Administration in collaboration with the Recreation Therapy Service under the direction of a certified recreational therapist, and establish a director of therapeutic service dog training. Veterans may participate in the program in conjunction with the VA's compensated work therapy program.