Instructors can also go on LOA. Normally when you go on LOA you just fill out a form listing all your students and who is going to be taking care of them. If your LOA is just for a few days, you still need to list instructors to take care of your students, but if your students don't mind taking a day or two off, they don't have to fly.
For Christmas, you have the choice of 2 standardized LOAs. You can leave a little earlier and be back before the 1st of the year or leave a little later and have New Years off. In reality it doesn't really matter, since almost all of the students leave for 2-3 weeks. This is the only time you don't have to deal with finding someone to take care of your students. If a student wants to fly over the holidays, an instructor who is staying in town is assigned to them.