I'd say probably not. I think they'd share gifts with their humans, with the implication being that the people who gave the gifts were helped out by their daemons in picking them, so sort of gifting that way.

This makes sense to me. Of course now I'm imagining the sorts of gifts Howard and Maria exchanged with the help of their daemons.

That said, obviously Santa has a daemon, and maybe she brings little treats for the daemons of the good little boys and girls?

To go with your theory, I'd imagine Santa's reindeer daemon simply helps pull the sleigh full of gifts.