Here's an old favorite. Although the song doesn't mention Christmas specifically, it's always felt like a Christmas song to me. So here it is, "Bells" by Wobble Test, my all-time favorite Milwaukee band.
This is from their debut cassette, from 1990, called trixienickybambibo (also known jokingly as "Four Dead Dogs"...since that's where the names in the title come from, each band member's childhood dog). That's my memory at least...someone can correct me if I'm wrong.
Wobble Test "Bells" (trixienickybambibo, 1990 cassette)