There are 2 state parks on the IL side. Delabar State Park north of Oquawka on the Keithsburg Blacktop about 20 miles north of Burlington. Nauvoo State Park is about 30 miles south of Burlington. Electric only. Sewer dump and water available. Both are on the Mississippi River.
There's not much around Delabar. Oquawka is the county seat but not much there other than a few bars and the burial site of Norma Jean Elephant. A circus elephant that was killed by lightning in 1972 and left by the circus so the town buried her near the center of town park. Use to be able to get some great fish at some of the bars in town. Other than that can't think of anything notable in the county.
Nauvoo is a bit more 'exciting'. The Mormons settled there in the early 1800s. At one time it was one of the largest cities in IL. The organizer of the Mormons, Joseph Smith, was killed just southeast of there in Carthage IL. The Mormons moved on and settled in Salt Lake City. They rebuilt their church as it was in the early 1800s. Pretty impressive. I got to tour it after it was built and before it was opened. Below the church and across the highway from the state park is the restored Mormon village with the homes of several of the Mormons including John Browning. Nice tour and free. Other than that Nauvoo isn't the metropolis it once was. Use to be the end of Aug/first of Sept was the Nauvoo Grape Festival which brought in thousands of people. Last time we attended it was just a small town festival. The state park use to be nice and clean altho I haven't been there for a few years. Had a nice view of the Mormon village and the Mississippi River.
Those are the closest to Burlington that I recall. I'm somewhat familiar with the Iowa side of the river and can't think of any campgrounds closer.
I worked that area years ago when I came "on the job". You'll know what that term means.
Camped in every state