Hi there, Mary!
Disney has created a mobile-friendly version of its
dining reservation Web site. Because it isn't an independent application, but a different "view" of the main site, you will not need to download and install any additional software to your smart phone. In fact, when you navigate to the dining site, you will find that the program detects your specific phone type and launches the compatible version automatically.
This optimized mobile site will allow you to find a table based on a number of criteria: time and date, cuisine, location (e.g. a specific restaurant, Park or Resort) and more.
There is no official downloadable app that records and provides availability for restaurants; the number of restaurants and the quick rates of change and turnover would require a great deal of updating, and that degree of realtime information can be pretty memory and battery-draining on a small mobile device.
Give the mobile site a try; it's fingertip-friendly and can head off a lot of hungry confusion in your day.
See you later, celebrator,
-Deborah B.