Tolkien was always careful to keep the specifics of his Christian faith out of Middle-earth, so I couldn't really let this be a Christmas card. Therefore I am interpreting this as a Yule card, a holiday the Shire folk celebrated.
Well, then, if it's not a Christmas card, who is the person in the Santa outfit? I've decided it's Barliman Butterbur, the portly, kindly, red-cheeked, ever-bustling innkeeper of Bree. (He must have had a few pints when this picture was made, otherwise he wouldn't have been able to sit still long enough for Frodo to perch there.) The red outfit is Barliman's Yule costume, the festive gear he keeps especially for greeting guests arriving in the depths of winter.
Of course, if you want to imagine that Frodo is sitting on Santa's knee making his Christmas wish, feel free. But something tells me that whatever Frodo wished for, it it would be for someone else, not for himself.
Here is the real image, an antique Christmas card I found on the Dover website:
I used this FOTR screencap for the face. I love his winsome smile: