Dear Marni,
Yes, restaurants an wineries will be open during Christmas time in Castellina in Chianti and all the other locations of the Chianti area. You may reserve your Christmas lunch or dinner (better reserving your Christmas meal in advance, there will be lots of people that day) or decide to enjoy the local delicacies on a different day, when restaurants will be maybe less crowded.
Greve and Strada in Chianti will be home to typical local Christmas Markets on the 8th and 14th of December the former, and on the 13th and 20th the latter. If you are round here, I suggest you take a look at these traditional events with music, handicraft products and obviously Santa Claus!
Enjoy your Tuscan time!
Chiara