Burlington has gotten some bad press recently due to homelessness and drug problems. It's not as bad as the press would have you think. Still, not great to look at though.
North beach is the only city campground I know of and its crammed into a spot near the old HS. You'd still need to take transport into the downtown area. I think you're better off staying in Grand Isle and making day trips. Champlain Islands are very nice. I don't know how avid a cyclist you are, or if you have e-bikes, but its a good area for that. There's a bike ferry across a small straight that connects the islands with the Burlington bike path. Vermont is small and largely rural so , other than in Burlington proper, you'd need a car to get around to see the sites. Lots of nice small towns to visit.
I lived most of my life there, now I look at it daily from my perch across the lake in NY.