In case you hadn't noticed, this is a season with which Reindeer (Rangifer tarandus) are much associated. In fact, the first part of the term, coming from Norse hreinn, actually means a deer, so it is a creature with a tautologous appellation. In North America it is sometimes called the Caribou, from Micmac qalipu.
The reindeer is, of course, associated with the drawing of the sleigh of Santa Claus on Christmas Eve. A set of names for Santa's reindeer was devised in the poem The Night Before Christmas, often attributed to C. Moore, though some doubt exists about the authorship. The poem is also entitled A Visit from St Nicholas. It appeared in the Troy Sentinel in 1823. The famous Rudolph is a more modern invention from a song written by Robert L. May in 1939.
But how were these reindeer supposed to be able to fly? The idea may go back to early shamanic practice. Shamans knew they could have the illusion of flying by consuming fly agaric mushrooms, but these have unfortunate side effects on humans. The shamans noted, however, that reindeer consumed them with impunity. The shamans therefore drank reindeer urine, which gave them the feeling of flying without any of the attendant discomfort. The Finns believed in Old Man Winter, who flew through the air in a sleigh drawn by flying reindeer.
Just to be on the safe side, I would suggest you don't drink reindeer urine yourself. I would imagine it's quite hard to find a store which stocks it, anyway.