Well, the law in Illinois requires you to report the incident, and the deer belongs to the person who hit it. In a perfect world, you could cell phone the game warden and he would dash up to the scene in minutes to dispatch it.
More than likely, it will take much more time for anyone to arrive. You have to also consider that in some counties, multiple incidents like this occur on almost a nightly basis. So, getting anyone other than a bored police deputy out there that evening might be a problem as well. However, I would make a reasonable effort to contact a game warden and get them out there. And by the time he arrives, the deer is out of its misery and my tire iron is back behind my truck seat. And the next day, the lady and I are grilling backstrap and sipping red wine....