Webster has a small campground which is on the east side of town, near the lake. It is closer to downtown but also smaller,
Google map, or
campground info. I'm not sure of they offer any utilities. We have driven thru but never stayed there, just too close to home.
Hamlin Beach is the other campground closest to downtown. If you stay at Hamlin Beach and approach it from the east on the Ontario Parkway, don't. There are several low clearance bridges that will take your roof off. There are other bridges on the parkway that are arched types. They offer full clearance, but only in the center. We avoid all parkways in NY & NJ because bridges are often too low for us.
The closest full hook-up campground is further away such as Southwoods RV Park in Bergen, or others.
Walmarts and Cracker Barrel locations that are options too. The south side locations are the better side if the city.