There are not any campgrounds that I know about that are close to the city, but there are a variety of choices within 30 minutes or so. Here's a good link - Camp, Lodge & Park an RV | Visit Buffalo Niagara.
There is a lot to see and do in the Buffalo / Niagara Falls area in the summer! It's a great time of year to visit. Here are a few links if you're looking for other activities.
You can find a few state parks both south of the city and north of the city in Niagara county. You can also find some state and private campgrounds east of the city near Darien and Darien Center.
We like Lake Erie Beach and Evangola State Parks.
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A very nice KOA on Grand Island. A tad expensive but close to Niagra Falls and Buffalo
But there is an Amusement Park right next door - be prepared for noise. It is small, called Fantasy Island. Very nice park to visit. Try FOUR MILE CREEK STATE PARK in Youngstown, NY. Lovely & Quiet!!
Evangola, State Park is on Lake Erie and closest to Buffalo, Smaller park with sites backing up toward lake, New shower/rest room easy access to city as it is just off route 5 which runs on the lakeshore,closer to city and is a main artery into the city,
The other mentioned parks are great to.
Enjoy your stay