Actually, "Santa Claus" was a song originally performed by The Sonics, one of the early bands in the Seattle music scene. Their garage sound influenced not only the later grunge sound from that city, but also bands from all over the world. Originally from a 1966 record label compilation, this song has been recently reissued by the still-active band.
Strangely, I also asked for a brand new car, a twangy guitar,
But you don't want to hear me prattle on about all this. You want some music. Here are They Might Be Giants, performing this great song, live on Good Morning America in 2009. Let's forget about them butchering the lyrics for a minute - and boy, they did butcher the lyrics. What they bring to this song is a cleaner sound, complete with horns, and that makes it a fun version.
In my opinion, the definitive version of this song was the one done by Belle & Sebastian. Clearly, the Welsh group are fans of the garage sound, because they retain it more..... even while bringing a flute to the show. Also, they know the lyrics. (My eight year old daughter knows the lyrics, too. It's her favorite carol)
The song starts at about 1:58 in this video, from a live performance, and it's a fantastic version.
As an added bonus, here is the original, which is just a terrific modern Christmas carol. This is what garage music is supposed to sound like.