I'm sure you've noticed Rend Lake nearby. There are tons of paid campgrounds though it looks like you have to go through the nightmare of recreation.gov reservations to find out about costs, etc. Maybe there is a number you can call to just ask.
There are also private campgrounds like KOA and a couple of Good Sam near Benton.
To the south is the Shawnee National Forest so you can look at the Forest Service site for that info. But if you have to be in Benton or Mt. Vernon each day, that is likely going to be too long of a drive. The forest has pretty parts to visit though they are not spectacular.
I hope you do not have to camp in that area past June...or you have air conditioning and are willing to camp at a site with electric. Heat, humidity and bugs (ticks, chiggers, mosquitos, etc.).