Depending on the season and the restaurant, you might get by without reservations, but if you're going to do it, do so with a "hope for the best, expect the worst" mentality.
Our last trip was April 29-May 6 this year (and we actually arrived a day early, April 28). For the duration of our scheduled vacation, we were also on the DDP. Because we had a little girl (3) and an infant, there were certain reservations we had to cancel (always within 24 hours) because we realized we would need more down time that day, and there were also times when we were planning a counter service meal but were so exhausted that we wanted to sit a while and cool off without fighting for a table or feeling rushed once we had it. We did walkups twice at The Wave, once for lunch, and on checkout day for dinner before heading home (to be honest, we wound up making reservations for dinner at the Wave that day, a few hours beforehand, but had we walked up, we would not have had a problem getting seated, the place had plenty of free tables). We did lunch once at Kona Cafe. And surprisingly, we were seated once for lunch in Liberty Tree Tavern with only a 10 minute wait.
Now bear in mind, we were not there during a busy period - it was after Spring Break and Easter but before schools let out for the summer. Your mileage can and probably will vary. There are safer restaurants to try to dine without reservations, but they're usually in the hotels and they might not be your choice. It would also be easier to make it a late lunch, as most people want to eat between 11-1:30, but most restaurants keep serving lunch until almost dinnertime.