Yes as a matter of fact. Originally he would write letters to his children as though they were from Santa himself. And they did follow a story-line. They were mostly about life at the north pole with Santa, the Polar Bear, the elves, and the occasional goblin war. This version of Santa was incredibly old but knew how to handle himself in a fight. These letters were later composed into a book, here's a look inside on amazon https://www.amazon.com/Letters-Father-Christmas-J-R-R-Tolkien/dp/0618512659
JRR Tolkien wrote Letters from Father Christmas to his children, telling them about battles with trolls and adventures with elves and all sorts of crazy things. For anyone that hasn't read the collection, it's pretty great. I don't plan on having children, but if I did, I would do this.
There's no specific site to find all of these, but there is a book on Amazon and...well here's a Google search.