Maybe and then again maybe not. I have gotten there on a Thursday afternoon (late April so still very chilly at night) and I took the last available spot. Why? A coming storm was bringing the AT hikers in for the night and then took up most of the campsites. I would try to get there on a Thursday to get a good campsite. When you drive in through the park gates they have a status board for all campgrounds so take a look at that and be prepared to go to Big Meadows or Loft Mountain which have a good amount of first come/first served sites
If it is a very busy weekend you might need to be there early on Friday, since it is first come first serve
All the sites are walk ups, so there shouldn't be a problem getting a spot.
Thanks! Your answer is awaiting moderation.