Bilbo was almost certainly intended by Tolkien to be heterosexual
In the very first draft of The Fellowship of the Ring, Tolkien wrote that Bilbo dropped out of Hobbit society in order to get married and have little hobbitlings - one of whom would grow up to become the protagonist of the book:
[Bilbo] said this very loud and everybody sat up who could. 'Goodbye! I am going away after dinner. Also I am going to get married.'
That's that. It merely serves to explain that Bilbo Baggins got married and had many children, because I am going to tell you a story about one of his descendants, and if you had only read his memoirs up to the date of Balin's visit - ten years at least before this birthday party - you might have been puzzled.
The History of Middle-earth VI The Return of the Shadow Part I: "The First Phase" Chapter 1: "A Long-Expected Party" (i) The First Version
This version was swiftly abandoned, but the third draft introduces Bingo Baggins, Frodo's literary forebear, as the son of Bilbo, and says that Bilbo had gotten married at age seventy-one:
When Bingo, son of Bilbo, of the well-known Baggins family, prepared to celebrate his [fifty-fifth >] seventy-second birthday there was some talk in the neighbourhood, and people polished up their memories. [...] [Bilbo] did two more things that caused tongues to wag: he got married when seventy-one (a little but not too late for a hobbit), choosing a bride from the other side of the Shire1, and giving a wedding-feast of memorable splendour; he disappeared (together with his wife) shortly before his hundred-and-eleventh birthday, and was never seen again.
The History of Middle-earth VI The Return of the Shadow Part I: "The First Phase" Chapter 1: "A Long-Expected Party" (iii) The Third Version
Bilbo's marriage would be abandoned in the next draft, with Bingo shifting to Bilbo's nephew, where he would remain. Christopher Tolkien remarks in his commentary on the chapter that he thought the abandonment of Bilbo's marriage to be "inevitable", but doesn't discuss why.
Regardless, the published Lord of the Rings contains no references to this original concept; at no point does Bilbo express any romantic interest in any character, male or female. His reasons for remaining a bachelor are discussed in Unfinished Tales, where Gandalf remarks (emphasis mine):
I guessed that he wanted to remain 'unattached' for some reason deep down which he did not understand himself - or would not acknowledge, for it alarmed him. He wanted, all the same, to be free to go [leave the Shire and have adventures] when the chance came, or he had made up his courage. I remembered how he used to pester me with questions when he was a youngster about the Hobbits that had occasionally 'gone off,' as they said in the Shire.
Unfinished Tales Part 3: "The Third Age" Chapter III: "The Quest of Erebor"
I'll grant that you can read this passage, particularly the first sentence, as a metaphor for deeply repressed homosexuality in a homophobic time. But I would say that the rest of the passage (coupled with Tolkien's early drafts) rather puts paid to that reading: Bilbo never marries because, subconsciously if nothing else2, he wants to be free to go have adventures.
It's worth noting that it isn't unprecedented for characters in Middle-earth to remain unmarried. Even among Hobbits, there are several with no children (Isengrim III springs to mind), which doesn't necessarily preclude marriage but does hint in that direction, and still more with no spouses or children listed in the family tree (such as Dora Baggins). There's at least one Hobbit, Ferumbras III, who is known to have never married; his mother was so unpleasant that he was allegedly unable to find anyone willing to cohabitate with her3:
Her son, Ferumbras, had no wife, being unable (it was alleged) to find anyone willing to occupy apartments in the Great Smials, under the rule of Lalia.
The Letters of J.R.R. Tolkien 214: To A.C. Nunn (draft)
1 Later in the draft, Tolkien reveals that Bilbo married Primula Brandybuck; interestingly, Primula would remain as Frodo's mother, though her husband was changed to Drogo, a cousin of Bilbo's
2 And possibly part of the design of Ilúvatar
3 A peculiar quirk of the Took clan is that the entire family lived in a single, massive Hobbit-house: Great Smials