I am planning to spend a winter in Tbilisi this December. I would like to know if the taxi is available 24 hours at winter? Is there any website where I can book a car/shuttle to bring me to the hotel from the airport?
Yes, taxis are available 24/7. A simple Google search revealed:
You can book at least one service online, but it appears to be just the equivalent of a paper form.
Perhaps your best bet is this: