If there was one thing L. Frank Baum didn't dwell on, it was romantic relationships, particularly between his main characters.
In his first three Oz novels, he tells us of two marriages, but they are fairly unimportant to the overall plot. (Quelala and Gayelette, Jinjur and her never officially named husband.) The most prominent romances were lifted from dramatic Oz tales (Private Files and Ozga from The Tik-Tok Man of Oz and Pon and Gloria from His Majesty, the Scarecrow of Oz). Throughout the series, we encounter several married couples both in and out of Oz, so we may assume that sexuality is a thing in Oz.
There's a few definitions of sexuality, but the one I refer to here is a capacity for intimate relationships. So, no, I'm not discussing people having sex in Oz. (Though Thompson did have Prince Pompa of Pumperdink and Peg Amy have a child, so it happens.)
One objection fans raised to the film Oz the Great and Powerful was that Oscar Diggs, the Wizard himself, was depicted wooing different women. (Actual womanizing was only suggested.) And to be honest, I didn't have a problem with it. He's the Wizard of Oz, not the Eunuch of Oz. That said, by the time L. Frank Baum's stories begin, he doesn't seem to have much interest in his sexuality. In all the Famous Forty, and in all of the stories set in the same continuity, the Wizard is happily single.
Also in all those stories, Glinda, Ozma, Dorothy, Trot, Betsy, the Shaggy Man and Cap'n Bill never seem to take a romantic interest in anyone. To be honest, a little bit in Jack Snow's "A Murder In Oz" revealing that Glinda has handsome young mountain giants serve her late at night amused me because it hints at her having a sexuality.
The most complicated case of sexuality in the Oz books is Nick Chopper, the Tin Woodman. He seems to be a typical heterosexual in The Wonderful Wizard of Oz, discussing how he decided to marry a girl that would be revealed to be named Nimmee Aimee in The Tin Woodman of Oz, determining to go back and marry her when he's received a heart. However, he doesn't do this. Instead, we discover in The Marvelous Land of Oz that he's inclined to be a dandy, and he and the Scarecrow share a "bromance" that begins to raise a few eyebrows, and by The Tin Woodman of Oz seems to be a bit more than that, as these two seem to be happiest with each other. Even more curiously, when the Tin Woodman finds his human head in The Tin Woodman of Oz, it appears to almost have a different personality than he does.
But doesn't the Scarecrow flirt with Scraps in The Patchwork Girl of Oz? Yes, he does. And in the Famous Forty Plus book The Runaway in Oz, Scraps becomes devoted to Popla, a plant that seems to identify as female. If these relationships are to be taken seriously, I don't think these manufactured people care about monogamy.
That brings us to the subject of homosexual relationships in Oz. (This means that both people in the relationship seem to or do love one another, and they are of the same gender, not that they've identified a sexual orientation.) Nathan DeHoff commented that given how many LGBT people write Oz stories, it's surprising that there aren't many openly queer characters in Oz stories. Isabelle Melacon wrote a piece for the Namesake website that suggests that such relationships would be a non-issue in Oz. Everyone lives forever in Oz, and are happy to be content and don't mind seeing other people content, so what would be the point of objecting to a homosexual relationship as long as the people in it are happy?
This isn't that there hasn't been Oz stories featuring queer characters. In Eric Shanower's "Abby," his adult Tom from The Shaggy Man of Oz has recently broken up with his boyfriend. In Chris Dulabone's The Fairy Circle in Oz, a vain king wishing to marry the most beautiful person he can find ends up marrying a man. Gregory Maguire has queer undertones in his Wicked Years series, particularly the bisexual Liir, but this isn't canonical Oz. Finally, in a story I recently finished for an anthology I was asked to contribute to, I took a shot at developing a queer romance in a traditional Oz setting. I hope I handled that one well...
Now, we must note that of course, Baum's books were written for children at the beginning of the 20th century. Sexuality of any type just wasn't a major plot point for children then. And anyway, Oz is more about the magic and adventure! But still, as we find characters forming relationships in the series, we will always wonder about the nature of these relationships.