I am thinking of visiting the Amalfi Coast for a few days between Christmas and New Year's day. Are the coastal cities of Sorrento, Positano, Amalfi, and Ravella "closed for the season"? I noticed that some of the hotels in these cities are closed between November and March. Will most hotels, restaurants, and shops be closed during this time of year? What about the Isle of Capri? Any information will be very helpful or sources of information. Thanks.

These are largely summer/beach resorts and are mostly shut down during the winter. You'll find a few places open, but not many. Capri in particular may be difficult to get to and from because of winter sea conditions. What are you looking for? Frankly, at that time of year, I'd stay in a city -- Naples if you want to be in the area.

Some people live in the Amalfi Coast towns year-round, but yes lots of hotels and restaurants will be closed. The big luxury hotels will have closed by about Nov. 1st, the ferry service along the coast suspended and the bus service reduced. Ferry service to Capri from Sorrento and Naples continues but at a reduced frequency that might make a day-trip unrealistic. Ferries can be cancelled if the seas are rough.

For more lodging and dining options and overall activity, you'd be better off staying in Sorrento. If it just rains and rains (definitely a possibility), you'd be able to go to museums and other sights in Naples. There aren't many indoor sights in the coastal towns. Rain or no rain, it will be dark shortly after 4:30 p.m.

The upside to all of this is that the coast towns are very, very quiet (except for Amalfi on New Year's Eve when there are fireworks) and getting a seat on the coastal bus much, much easier.

As said above, local people lived there all the year round and buses ran all the year round before mass tourism broke out and they will continue to do so.
I suppose you know what to do in the case of cold, windy and rainy weather.