It is definitely doable bluey. Might not be easy, though. One of the best bets would be Idaho. You can buy over-the-counter tags for mule deer, white tail deer, elk, black bear, mountain lion and wolf. Pronghorn would be a draw tag unless you go guided on the Duck Valley Indian Reservation which is a great option. ALMOST guaranteed success and good quality bucks. Last I knew it was $1500 for 1 buck and I think 4 or 5 days hunting. A close friend and business associate of mine goes every year religiously. One of his favorite hunts and he has taken some outstanding pronghorn.
The challenge is timing all three together. Pronghorn you're probably going to hunt July - September. Mule deer you could bowhunt in late August and September, but rifle hunting would probably be October & November. Mountain Lion opens August 30 but is a lot tougher without snow on the ground and fur is not near as good. Best time to hunt cats is December through February.
So, you could hunt all 3 three plus elk, bear and wolf in September, but Lion might be more of a long shot. You could hunt bear instead of lion and have a real good chance at getting all 3 especially if you had 3 weeks.
Most other states you are going to have to play the lottery for draw tags...