There are three laws that cover this type of situation; the first two deal with dispatching the injured animal, while the other covers the disposition of the animal.
41-1-12. Euthanasia of animal injured in motor vehicle accident. Notwithstanding any other provision of law, any person who has seriously injured a wildlife animal in a motor vehicle accident or any person who comes upon a wildlife animal that has been seriously injured in a motor vehicle accident may euthanize the seriously injured animal so that it will not suffer needlessly if that person has the means, skill, and will to euthanize the animal humanely.
41-1-13. Report of euthanasia to department. Any person who euthanizes a wildlife animal pursuant to § 41-1-12 shall report the event to the Department of Game, Fish and Parks or any conservation office within twenty-four hours.
41-1-5.7. Disposition of deer and antelope killed by motor vehicle. If any deer or antelope is killed by a motor vehicle on a public highway, any person who desires to possess that animal shall notify a conservation officer, as defined by § 41-15-10, prior to taking possession of the animal. The conservation officer may give a dated and written authorization allowing possession of the animal. The conservation officer may give verbal authorization to take immediate possession of the animal prior to receipt of the written authorization. The written authorization allows the possession and disposal of the animal. The written authorization shall remain with the carcass while in processing or storage. No part of an animal so obtained may be sold, bartered or traded. There is no fee for the issuance of such authorization.
Please contact your local conservation officer for additional questions or for more information.