You're likely going to be going through Cody, Wy from Illinois to the east entrance if you're going to stay at Fishing Bridge. There are several rental companies based at the airport there in Cody, which wouldn't be too far out of your way. Unless you were planning on some other route, otherwise it looks like West Yellowstone has the corner on the market as far as proximity to the park, but then you'd have to traverse the entire park to pick up the car.
Hertz has a rental location in Cody, Wy and you could drop off at Jackson, Wy, and Thrify does too, but their website is being wonky.