RV/Travel Trailer Rental?

I"m assuming you mean you're wanting to rent a motorhome or a travel trailer (both are RVS). One of the largest nationwide Motorhome rentals is Cruise America but I don't believe they rent trailers. There is a lot more to renting a trailer considering the adequacy of various tow vehicles, hitch installations and driver skill sets required for towing, but stopping at most any DEALER who sells these may be able to assist you in finding someone who will rent them.
Here's a link to Cruise America
http://www.cruiseamerica.com/

To rent locally there is Freedom RV in Rapid City, also Cruise America thru Black Hills RV, and Jack's Campers.

as was said, TT's are RV's.
and ditto on needing to have more info on your tow vehicle, in regards to renting a TT.
that's why you see all kinds of MH rentals but not a lot of TT's for rent.
you need a tow vehicle that has a towing package, enough payload capacity and usually a WDH with sway control.

as far as a private owner rental, don't look here for someone, since nearly everyone here would never rent their RV to a stranger, let alone friends or family.
there are tons of threads here about letting someone use your RV. over 90% would discourage it.
you'll be better off finding a MH private rental than a TT.

So you're from Kentucky but you want to rent an RV in Rapid City, South Dakota? Maybe for the Sturgis Motorcycle Rally Aug 3-9? If so, the folks that live there know what a patch of dirt the size of a sleeping bag is worth that week.
If not, you're going to find the area really crowded and expensive then.

Just saw another post about RVshare.com. That may be an option for you. It appears that they are an intermediary for owners and leasers to connect. They mention insurance, too. The owners may be private individuals or companies.

I can't compare against others but I can give you a single data point.

We rented a popup from Jack's last year to see if we wanted to move away from tent camping and to save ~1K towing miles vs. renting locally. We then proceeded to put over 2.5K miles on it staying in the Badlands and Custer before heading out to hit several national parks in Wyoming, Montana, Utah, and Colorado.

The company was easy to work with when reserving, though the pickup process was a bit convoluted as the main rental guy was out of the office on the pick-up day and the fill-in read the reservations wrong so he gave us the wrong unit (same model) which had not been fully prepped prior to our arrival. When we returned it we heard "Oh, we've been looking for that one for weeks".

Overall it was a positive experience and I would consider them again in the future.

peer-to-peer rental website rvshare.com shows 6 pages of rentals Rapid City in July.