Wounded Warrior Service Dog Act of 2013 - Directs the Secretaries of Defense (DOD) and Veterans Affairs (VA) to jointly establish the K-9 Companion Corps program for the awarding of grants to assist nonprofit organizations in establishing, planning, designing, and/or operating programs to provide assistance dogs to certain members of the Armed Forces and veterans who have certain disabilities. Defines "assistance dog" to mean a dog specifically trained to perform physical tasks to mitigate the effects of such a disability, except that such term does not include a dog specifically trained for comfort or personal defense.