Yes, you can hunt it with a muzzleloader, archery or shotgun. I know the rules state no lead or toxic shot, so you may have to look at a copper bullet or non-toxic one. Given it says shot. I just use the Hevi-metal shells in the shotgun during modern typically.
One thing to pay special attention to is that they transferred the Columbia whitetail and those are protected. They transferred them to the Ridgefield refuge and whitetail typically don't range to far, but you never know. It's a short distance as a crow flies.