Wounded Warrior Service Dog Act of 2019 This bill provides for the establishment of program for the provision of service dogs to certain Armed Forces members. It requires the Department of Defense to establish a K-9 Service Corps Program to award grants to nonprofit organizations to assist them in planning, designing, establishing, or operating programs to provide assistance dogs to members with a disability who are receiving receiving medical treatment, recuperation, or therapy; those in medical hold or medical holdover status; or those on the temporary disability retired list in cases where medical principles indicate that the disability may be of a permanent nature.