Good question.. Black Hills has a ton of gun people but you never hear about where they shoot.. In the hills it's not quite like it is out on the tundra (Buffalo County) where you just go out in the country and "Start Blastin"..
Id call the chamber of commerce and ask.. they'll probably fall all over themselves to find you a place to go shoot or set you up to tour something gun related like Dakota Arms or Corbon.
Deadwood Chamber of Commerce & Visitors Bureau
767 Main Street
Deadwood, South Dakota 57732
1-800-999-1876
Spearfish is a couple miles north, Spearfish Rifle & Pistol Club has a 200 yard range. I've never shot it. 605-642-4093
If you want to wander all the way to Rapid City the National Guard range it's kind of open to the public, they have a 300 yd rifle range.. I've never shot it.. Try 605-399-6703 and see what happens..
Call 605-348-5150 and see if you can tour the Black Hills Ammo in Rapid.
Call 800-626-7266 and see if you can tour the Corbon facility in Sturgis.
Call 605-347-4686 and see if you can tour Dakota Arms in Sturgis.
Hope that helps..