You'd need to ask for at least three grammatical cases: nominative, accusative and possessive.
We considered trying this a while back with our organisation's personnel tracker, but concluded it needed a disproportionate amount of user effort. So we abandoned it in favour of just stating the user's name and/or singular "they" everywhere (in line with UK GDS guidelines[1]).
Which is partly why I find the pronoun database at "Pronoun Island" interesting/useful because its a database of most of the common substitute pronouns and almost all of their English grammatical cases.
Though, as you determined singular "they" is a simple enough implementation without needing a database lookup and keeping a pronoun database on hand.
[ETA: It's a small shame this particular DB is AGPL, keeping it from being maximally reuseful as a shareable standard library. Though data is not code and AGPL may not technically apply, but there's no separate data license such as a CC license mentioned in the repo, so one must assume AGPL-like conditions.]
Great resource, thanks! It would be interesting to see more context for each of the entries, such as examples in published work—I certainly noticed they'd incorporated ones used by science fiction authors that I like. Though I understand that such examples may still be quite thin-on-the-ground.
[1] https://design-system.service.gov.uk/patterns/gender-or-sex/