i would ask the taxidermist if they want you to skin it. some animals are skinned in different ways depending on what position the mount is going to be. the position helps hide the incision.
if anyone happens to have a squirrel mounted climbing up or down a tree take a good look at the belly side, use a flashlight if you have to. some of them arent even sewn shut, just stapled to the form and you dont even know it as the belly incision doesnt show. just like a one sided wall mount fish, ever see the back, most arent finished, just stapled and closed over with bondo then painted black and only one eye installed on the show side.
if you skin the coyote the wrong way, it could be worthless to the taxidermist. ask first, maybe the taxidermist, just wants it whole.
also, if you can not get it to the taxidermist right away, do not leave it lay around, get it home and in the freezer, do not place it in a plastic bag until it is cool, the bag will lock in the heat and cause it have some hair slippage.