O.K. Usually South Dakota is the destination for prairie dogs BECAUSE the Native Americans have Reservations where, among other things, they market outdoor recreation including shooting pasture rats. No reason they couldn't help you with coyotes too. (Wild dogs). Rosebud is south center. S.W. Pine Ridge. Central, Brule Sioux. North Center. Standing Rock, Norther Cheyenne. If you google them, you can get the latest offerings, prices, etc. Costs are lower in Wyoming, but you said SD.
And there are many, many guides that offer hunting services. I recall on on lakotamall.com.
What you need to bring depends on how much you want to spend. Spend more, bring less.
To put a point on this, you may be over 150 miles from a McDonalds. Food? Yes, there are diners in towns, but towns may be 25 or 50 miles apart. This is not the place for some 25 year old pick up that may or may not last another year. AAA doesn't leave paved roads to retrieve... 2nd spare is not a bad idea. Lots of water if you are going "off alone." GPS? HAPPY HOLIDAYS. lucik.