Weekends in the summer can be pretty tight everywhere in Canada.
Our summers are too short!
Unless you can schedule your visit Mon-Thurs, I'd reserve.
Dunno about lead times though. Maybe pick a few sites and call?
We usually stay at Parc Nationale de la Jacques Cartier about 30 min north of Quebec City. Depending upon what you're looking for, they may have a few mid-week spots available.
https://www.sepaq.com/en/reservation/camping/parc-national-de-la-jacques-cartierEDIT: Just saw your RV. Prolly the only thing Jacques-Cartier would have is an unserviced parking lot. But it's still a beautiful place. We've camped in their parking lot and had a great time.