K-9 Companion Corps Act - 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 nonprofit organizations to assist them in establishing programs to provide assistance dogs to certain members of the Armed Forces and veterans who have certain disabilities. Defines the term "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.