Ah, the ever-present debate: Is singular they/them grammatically acceptable?
Short answer: Yes.
And now for the longer answer.
It’s a bit more complicated than this, but use of singular they has been around for centuries. The context is different from how we use it now, meaning that probably very few if any English-speakers were using they as a personal pronoun. It’s safe to say, though, that this contemporary use is born from the long history of using it when gender is unknown or irrelevant. For example, “Someone left their umbrella here the last time it rained."
Some of us (i.e., me) grew up learning that this is entirely incorrect and a person must alter the sentence so as to avoid pronouns or make the entire thing plural. For example: “People keep leaving their umbrellas here when it rains” or “I found an umbrella someone left here when it rained."
It probably comes naturally to some people to change the sentence, but I would guess most of us have to consciously rearrange things in our heads. That seems cumbersome and unnecessary, so casual use of singular they in speech is absolutely fine and correct. In formal or business writing, it might produce tighter sentences with a particular feel to them. As an editor, I wouldn’t automatically correct use of singular they, depending on the context. However, people should do as they see fit in that situation. (See what I did there?)
But what about in fiction? Some people would see it as stickier there. Again, as an editor, I would not change it. Use of singular they in fiction has a rich history, and I see no need to fix anything. Unless the sentence is overall poorly constructed, it shouldn’t need to be changed. Part of writing fiction is speaking to the language of readers. That involves a lot more than nitpicking an individual sentence where singular they is a placeholder for an unknown person or irrelevant gender. The majority of readers don’t complain about this use.
Which brings us back to use of singular they as a personal pronoun, this time in fiction. Obviously, in non-fiction, a person using it as their pronoun should be respected, whether or not someone else has an issue with it. In a story, the author is in control of which pronouns to use. So what then?
A not-small number of readers of queer lit have complained (some of them loudly) about how “confusing” it is to read singular they. I find it odd that people don’t seem to get confused about other pronouns when there are multiple people in a relationship using the same one. The majority of authors are careful to construct sentences to make sure everyone knows which he, she, they, or combination thereof is being spoken about.
This is mainly an issue of readers’ discomfort. Whether that stems from a lifetime of having it drilled into our skulls that singular they is bad or (more likely) a lack of understanding about non-binary genders, it’s still on readers to adapt. Times have changed, and it’s good for authors to challenge long-held beliefs.
So I say, go ahead and use singular they. If you need more convincing, have a listen to Bill Nye’s podcast on language with guest Benjamin Dreyer, the Copy Chief at Random House. The relevant portion on singular they is at around the 22” mark, but the entire podcast is worth the 50 minutes or so.