Not to get too deep into the weeds here, but this has gotten me thinking on this on what is probably far too ridiculous of a level of detail.
So first of all, to define terms, "sex" is what the body is right now, "gender" is the psycholodgical identity of the elf. I can see sex falling into at least four categories- male, female, and at least two categories or hermaphroditic/dualsexed, with the possibility of an asexual body type as well in some cases (though I don't think there is any canon support for that). But I think there's a strong likelihood of more levels of differentiation in elven bodies, and therefore in pronouns. There's also a chance that different pronouns exist for elves who shift sexes and those who don't.
Elven gender might not even really exist as a construct. I have always seen both elven gender and sexual preference as, typically, highly fluid; so if there is such a concept in elven society, it's probably very well-developed and detailed, accounting for lots of variants. So for instance, the pronoun you use for an elf might vary for an elf who changes gender when it changes sex, vs. one whose gender remains male regardless of its sex.
This is turning into a deep ramble... but yeah, I could see dozens, maybe hundreds, of different pronouns in Elvish. Of course, I also have the view that non-elves typically never learn more than advanced babytalk in Elvish due to its complexity; the long-lived elves have plenty of time to elaborate their language in ways that require deep experience to untangle. So a human who spoke Elvish would probably only use the most basic three or eight or something pronouns.