Sorry to hear about the poor fawn!
Two years ago I had a deer that was coming to my yard with a broken leg. I called our local rehab facility and the advice they gave me was that they won't do anything to help an injured deer. Of course, the deer around here are not domesticated...I'm afaid I don't know much about domesticated vs. wild deer, especially since it doesn't sound like the domesticated deer are really tame or anything.
Anyway, the reason the rehabilitator said they don't do anything is because trying to catch a deer is so traumatizing to the animal that they feel the deer could do more harm to itself. This was a rather large, adult deer...it came around for a few days and then I didn't see it anymore so I don't know what happened to it. I hope you have good luck with the fawn though...keep us posted!