Introduction: Father Christmas, St Nick or Santa (whichever name you prefer) has a lot of mystique - some may even say magic - about him. It is entirely within the realms of possibility that he could be the only wizard known by the vast majority of Muggles.
1) Travel.
Many Christmas songs have lyrics stating some version of “Santa went down / up the chimney”. Flooing is a wizarding method of travel that involves a fireplace. It may be the least likely option, due to the sheer amount of Floo powder that would be used, the hassle of connecting EVERY house to the Floo network for one night, and the cooperation needed between every country - some of whom must have been at war with each other at various times.
The other method would be Apparition, though this would only work if limitless Apparitions can take place within a short amount of time without depleting a wizard’s magical reserves.
2) Santa’s elves.
Depending on which Christmas movies you believe, they make the toys, check for naughty children, or help deliver the presents. Therefore, Santa (or whatever his name is) could be a wizard with hundreds of House Elves in his service.
3) Flying reindeer.
Why not? In a world that has snakes that can kill at a glance, birds that deliver the post, and sentient twigs living in trees, it certainly isn’t beyond the realms of possibility that reindeer couldn’t fly.
4) Santa’s sack.
It contains all the presents for every boy and girl in the world. How is that possible? An Undetectable Extension Charm. If Hermione can carry survival gear in a small beaded bag, and Newt can hold a menagerie in his briefcase, Santa can bring presents in a sack.