Could I use the gender-neutral “their” for the antecedent “account”?

Actually yes. As strange as it may sound, "account" refers to a person, in this instance. If you use Twitter, for instance, you will frequently see users use the word "account" as a synonym for "user." To convince myself of this, I searched for the phrase "one of the worst accounts," which was the first phrase I tried that sifted out other meanings of "account" ("this account is," for instance, turns up mostly remarks about newspaper stories). Here are some examples I found (no offense to the users in question; I do not know who any of these people are and have no opinion about their Twitter presences):

Mark Jackson Burner is the WORST burner account on here and is genuinely one of the worst accounts on Twitter dude. He is just really unfunny I can’t remember ever laughing at a tweet of his

https://twitter.com/Schnuzel/status/1419749671908544512

I can’t believe some of my mutuals follow this MF. This Easter Island looking ass dude is one of the worst accounts on here. Know he’s a Grade A weirdo.

https://twitter.com/PersnicketyK/status/1437426989841985545

Just want to add to this excellent thread that Rex Chapman is one of the worst accounts on this platform and RT-ing him is the kind of thing you may be asked about at the doors to the afterlife.

https://twitter.com/BDSixsmith/status/1419364909763244040

I found some more with "this account uses." Still everyone's pretty negative, but again, not familiar with the people being criticized:

Not really. The way this account uses it it's, "Get mad at what we post unless it turns out to be fake, in which case it's satire."

https://twitter.com/nate_e_u/status/1465063962307923974

@twittersupport this account uses your service to [harass] underage women and men [with] their photos, doxxing and threatening them

https://twitter.com/Hmmmmrestricted/status/1465025227306545165

If we accept that "account" has the meaning of "user," then clearly it may be used with possessive pronouns too -- "some accounts constantly insult their own followers," for instance, would logically be a sentence that I might write with this meaning.

If I might speculate on how this usage came to be, it's normal on Twitter for there to be accounts which are operated by more than one person. It would seem strange to refer to the Nestle Twitter account as a "user." But most Twitter accounts are just an individual user, so once people started writing things about what they thought about "accounts," or what "accounts" say or do, it was a natural next step to start applying that language to individual-user accounts and conflating the person and the account.