If you are not going at a really crowded time, walk up or check in the morning when you know what park you are going to be at. About 10 months ago we walked up the Tokyo Dining, Biergarten, Yak and Yeti, Rose and Crown, Plaza (checked a couple hours in advance on this one). We had a few reservations - ones on the way in or the way out. In the old days 5 or more years ago we walked up to everything inlcuding the castle (it wasn't a character meal then ) and other more popular restaurants. We would stay at the beach club and always walk up to Cape May for breakfast and dinner. It is different now - but it is still possible to get nice meals on the day of -- at least in my experience with our party of three.
