I haven't been to the WP walk-in sites for a couple of years, and in that time a 2nd area was built with more sites near the overflow parking. However, the few times I stayed, it didn't appear to be any strict enforcement of the length of stay. And I always found a spot, even on weekends.
I would say mid-morning to mid-afternoon would give the best chance of finding an open spot. Be a good neighbor, be careful with the food to minimize bear encounters, and be sure to bring ear plugs if you are a light sleeper. While the location of the walk-in sites are great for being so close to the TH and the portal store, it is not a place I would recommend for peace and quiet. Dayhikers leaving the campsites at all hours of the early morning, along with other dayhikers driving up, there is lots of activity.