Jared Davis has been discussing various Santa Claus stories written by the Oz authors at his own blog, and there isn't much I want to add to most of them. I do, however, want to say a bit about Ruth Plumly Thompson's The Curious Cruise of Captain Santa, which Jared discussed here. As he said, there's no direct connection to Oz, and no attempt to tie in with with L. Frank Baum's conception of Santa. His is sort of a red-clad, gift-giving Buddha; while Thompson's is more of a kindly and sometimes absent-minded eccentric. Also, Thompson's Santa lives at the North Pole (the location of the magical Christmas Country), rather than in the Laughing Valley. Still, if you wanted to tie Captain Santa in with Oz, I suppose Santa could have changed somewhat over time, and maybe decided to move his operations. The immortals of Baum's universe do tend to get restless from time to time, after all.
While I'm on the subject, here are some Oz-universe Santa and Christmas stories not by official authors:
"Santa in Oz," by Tim Hollis - Appearing in the 1986 issue of Oziana (the first one I ever owned), this is a humorous story in which Santa temporarily moves his operation from the North Pole to the Emerald City. Hollis' Santa mythos mixes Baum's take with the more modern interpretation from the Rankin-Bass animated specials. Santa does mention that he used to live in the Laughing Valley, but he's married (which neither Baum's nor Thompson's Santa was), and both Rudolph and Frosty make appearances. Tying this in with the canon is a little difficult because it has Ruggedo living in the Emerald City, in disgrace to the point where he's forced to work at a miniature golf course, but he's still largely allowed to move around freely. Still, it's a fun story that's worth a read, and it has an amusing running gag of a confused Jack Pumpkinhead asking Santa about colored eggs.
The Enchanted Gnome of Oz, by Greg Hunter - Another Christmas adventure with Ruggedo in a major role, which I guess makes a certain amount of sense, as Dorothy mentions his resemblance to Santa back in Ozma. Actually, Santa himself only makes a cameo appearance, but it DOES take place on Christmas Eve and Christmas Day. This tale also fits better into the canonical saga of Ruggedo than the last one, as it starts with a goblin magician restoring him from the cactus form he was given at the end of Handy Mandy.
Christmas in Oz, by Robin Hess - This one also ties together Santa's Baumian home with the now better known polar location, in this case by saying that he has homes and workshops at BOTH places. The story revolves around his setting up a third in Oz itself, specifically in a valley between the Quadling Country and the Emerald City territory. The plot of the story is pretty slight, but the additions to the Santa/Oz mythology are enjoyable.
A Silver Elf in Oz, by Marin Elizabeth Xiques and Chris Dulabone - Two American children seek out Santa to try to get his help in saving their town, and they meet up with a rebellious elf who's trying to go freelance. (Kind of similar to the plot of Santa Claus: The Movie, actually, but who else actually remembers that?) I haven't read this one in a while, but I remember the final resolution being kind of hard to swallow, especially in light of Ozma's remarks at the end of Tik-Tok. I won't give it away, though. I believe Chris and Marin have written a few other Oz books with Santa as a significant character, but I haven't yet read them.