There probably aren't going to be many people here who will have accurate information. The on-site restaurant is only a few months old, as it didn't exist prior to the recent renovation. According to the hotel website, there is no buffet.
My advice would be to call the hotel during normal business hours. The best time to call would be mid-day, as that is when it will be quietest. Ask to be transferred to the restaurant. They will provide the best information.
407-827-2727
Contact information:
http://www.buenavistapalace.com/contact-en.html
Depending on what you consider to be "walking distance", you can walk to the Marketplace area of Disney Springs. Both Earl of Sandwich and Wolfgang Puck Express serve breakfast. They are fast-casual dining, so you order and the counter and get your food once it is made. Personally, I think Wolfgang Puck Express is far better than the Earl. The normal opening time for both restaurants is 9am. However, on a busy week around Easter, I wouldn't be surprised if they opened sooner.
From the main lobby of the Hilton Buena Vista Palace, it's about 2,000 feet (0.37 miles) to those restaurants. That's roughly 8-10 minutes.
This is a recycled graphic I created for an older thread.
A: Main Lobby
B: Self Parking
C: Theme Park Shuttle Stop
D: Island Building
E: Pedestrian Bridge to Disney Springs
To get to Disney Springs, you can't cross the street. You have to go all the way to the pedestrian bridge. You can see Earl of Sandwich on the map. Wolfgang Puck Express is the next building to the left, where the fork/knife icon is shown.