They signed at Tusker House for breakfast, Akershus for breakfast and 1900 Park Fare for dinner in early May. I'd guess the only limitation is if the character isn't able to grip a pen (think Buzz Lightyear or King Louie).
The characters sign autographs and take pictures. Make sure you have your pen and book ready (right on the table). They move pretty efficiently through every restaurant.
Autographs are a great way for shy kids to begin their interaction with characters. As PP said, Sharpies or large pens are best. Don't wait to buy pens at WDW- $7 for a larger one with a decorative thingie on top. Even a pack of reguar Disney pens was about $9