Hallmark cards, and the Hallmark Hall of fame ( since about l952) have been around so long it is impossible to generalize on the Christmas Music angle alone.
We Wish You a Merry Christmas
The nutcracker song.
The song "Coming Home for Christmas" is featured in the Hallmark Channel movie called "Coming Home for Christmas."
Christmas Wrapping by the Waitreses
The 2011 Christmas commercial that Time Warner is airing contains a song by Guster, "Tiny Tree Christmas ♫ ... and then appears a perfect Christmas time"
Tiny tree Christmas
lose yourself
Raveonettes
Pomplamoose
Stevie Wonder - What Christmas Means To Me
This is clearly a song created by Kmart for it's 2011 Christmas Ads. It is simply a jingle, not a song.
Hallmark does have many Christmas-themed Thank You cards. They can be located by looking on the Christmas section of the Hallmark website, then selecting Thank You.