First off, let's refresh your memory with a replay of the BBC's Christmas Ident starring David Tennant as the Doctor.....
It's cute, you probably thought, but it can't possibly be considered as part of the official canon for 'Doctor Who'.
I say, why not?
Toobworld Central accepts that Santa Claus is an established character in the TV Universe. The real Santa Claus has shown up in the main Toobworld of Earth Prime-Time in 'Bewitched', 'I Love Lucy', and several others. He has been spotted in several other TV shows that take place in alternate TV dimensions, like that 'Twilight Zone' episode of "The Night Of The Meek". And at least one of the TV adaptations of "Miracle On 34th Street" (either the one starring Thomas Mitchell or the Ed Wynn version) deserve to be considered for the main Toobworld. (I'd send the Sebastian Cabot TV movie off to an alternate dimension.)
And if Santa Claus exists in Toobworld, then so do his flying reindeer. Even if these ungulates seen in the Christmas Ident aren't the famous members of Santa's sleigh team, they're part of the special herd that Santa breeds at the North Pole.
All the Doctor had to do was hitch these beasties up to the TARDIS and then guide them to pull it out of the snow drift. After that, their basic nature to haul a sled through the air took over and the Doctor got a special ride this past December.
As to why he was at the North Pole, I guess we have to assume he was visiting Santa Claus and perhaps even helping him out with some alien invasion of the workshop. Demonic toy attack, maybe? Whatever the reason, maybe there's a fanficcer out there who will supply the story....