Can you spend the night in your RV at the visitors center?

No

I'm not sure about that. There was only a parking lot with no hook-ups of any kind, and no outside restrooms that I saw. It is in town and not a busy place after hours, unlike some places. I would think not but I could be mistaken. I know that a few Walmart stores do allow self-contained RV's to park on their lots but... The trolleys meet their riders there and the riders pay to park their vehicle for the duration of the trolley tour. I suggest you call the visitor's center just to be sure.