You can probably find a spot
Exit glacier is right down the road. For the price $81-$76 bucks a night they better throw in a massage. The facility is nice but I decided $20 in a city campground was a way better deal. But in a pinch if you drove here and all others are full it is an option.
Exit glacier is a must. There are street signs showing where the glacier was in past years. Watch your speed because the Ranger will give you a ticket.