I used to live in Ketchum; wish i still did....
There are a few options. There's the Wood River gun club, down by Bellevue, you'll see it to the left as you drive up 75 to ketchum, before Bellevue. They have a rifle range. Then there's the Hailey Shooting Range, to the East of Hailey, which is rifle, pistol and clays. I think it may be the "Blaine County Range" but could be wrong.
Closer to Ketchum there's the Sun Valley Gun Club, which is clays.
Beyond that, take a look at a map. You can shoot on BLM land, and there's a lot of it. What i used to do was the head out Warms Springs Rd., about a half a mile after the hot springs, (which makes a great apres-shooting soaking location) and find a spot with no one around. There's stretches where you can find good yardage... just go up the road enough not to frighten anyone having a dip in the hot springs.
After that, get a beer and a prime rib at the Pioneer Saloon... and check out the crazy taxidermy and punt guns they have on the walls.
