Support Our Military Working Dogs Act This bill addresses provisions regarding the care and study of military working dogs. First, the bill authorizes the Department of Defense (DOD) to accept donations for the purpose of providing long-term care for retired military working dogs that have been adopted. The bill authorizes DOD to enter into a multi-year agreement with a veterans service organization or nonprofit entity to solicit, accept, and transfer funds to provide long-term care for adopted military working dogs. The bill prohibits charging a fee for the adoption of a military animal. The bill updates the duties of the Joint Trauma Education and Training Directorate. Specifically, the bill requires the directorate to develop a comprehensive trauma care registry with respect to military working dogs. Additionally, the directorate must inform and advise the conduct of research on the leading causes of morbidity and mortality of military working dogs in combat. Finally, DOD must establish a personnel management plan for veterinary care.