I believe you have slim to no chance of walking into a park restaurant on New Year's Eve. You might want to try some out of the way restaurants in resorts, but on that particular night even those might be full, and if you want to be in a park later you might want to consider whether the parks will close for capacity. Decide if you want to be in a park for the day - if you do, keep trying to get any sort of reservation there - people do cancel.
If a theme park is closed for capacity, you will not be allowed in even if you have a restaurant ADR. Magic Kingdom is the most likely park to close for capacity. If you intend to spend the evening there, you probably need to be there early and stay there. If that's the plan, eat counter service and be prepared for lines.