I think Bombadil was originally conceived by Tolkien outside of the Middle Earth and Silmarillion mythology. I'm not real convinced that Tolkien ever very specifically defined who he was.
I seem to remember Gandalf, in describing his unsuitability as an ally, saying that ultimately Bombadil would fall to Sauron as Sauron is able to torture even the earth. This seems to imply at least a connection with the earth itself.
Gandalf said something along the line that if all other resistance failed and Tom Bombadil was the only one left standing, he would be unable to defy Sauron. Not surprising, I think.