If the pelt is not a concern, pretty much any gun you want to pull the trigger on is fine. I've seen people with 7mm mags that shoot songdogs with them.
But if pelts are a concern, then the smaller the hole the better. I use a .22-250 with Remington Accu-Tip ammo and if shot strait in the chest, almost never exits. Even in a broadside shot I rarely get an exit wound that needs to be fixed to sell the pelt.
The .204 Ruger is a hot item for furbearer hunting as I've heard lots of good things from coyote hunters with them.
The difference is mainly in bullet construction. Varmit bullets all but explode on impact, minimizing penetration. Deer caliber bullets tend to maximize penetration to get to vitals in bigger game. Power levels aside- a .30-06 deer round will blow completely through a coyote, and varmint rounds will not penetrate reliably on big game- it's a matter of matching the tool to the game.
If the 7mm-08 (a round I don't have much experience with) has varmint bullets in the line up- then I don't see why not. But deep penetrating deer bullets are way overkill on coyotes.
Again- no biggie if you don't care about fur. But if you want to sell the fur later- try to put the smallest hole in it you can.