Sappho's poetry seems to place her in a circle of women, but it's unclear what exactly the nature of that circle was.

Holt Parker's 1993 paper published in Transactions of the APA is available for free online. It's the most thorough investigation I've come across of the "Sappho ran a school" legend/interpretation.

(1993 is ancient in scholarly terms, of course. But while we've turned up entire new worlds of Sappho's own writing since then, the historical sources of biographical data are still the same, much later tales.)

It depends how you understand "school". If you take it as a literal institution where girls go wearing pleated skirts, carrying a bundle of books, and taking notes from a teacher, then no, it is not true. That's the image referred to in this bit of the New Yorker article -

The eagerness to come up with “innocent” explanations for the poet’s attachment to young women persisted through the late nineteenth century and into the twentieth. The most tenacious theory held that Sappho was the head of a girls’ boarding school, a matron whose interest in her pupils was purely pedagogical. (One scholar claimed to have found evidence that classes were taught on how to apply makeup.)

However, it can make sense to talk of Sappho's circle as a "school" for certain understandings of the word. Later on in the article, where Mendelsohn says this -

Ancient references to the poet’s “companions” and “students” have led one expert to argue that Sappho was the leader of a female collective, whose role was “instruction leading to marriage.” Rather than expressions of individual yearning for a young woman, the poems were, in Lardinois’s view, “public forms of praise of the general attractiveness of the girl,” celebrating her readiness for wedlock and integration into the larger society.

In that view, the Sapphic circle would have run on a model comparable to that of a male hetaireia or of a sympotic society. That model is encouraged by the fact that many male poets of the time were celebrated especially in the context of the aristocratic male symposium. Among other things, the symposium had an educational function: it bound male social groups culturally and politically, and was a forum for personal and social acculturation as well as entertainment. The male symposium was not just an excuse for a party, or a men's club: it was also a kind of prep school for young adults learning to become members of cultured society. And this didn't revolve around memorising facts about geometry: it was a combination of social conduct, how to eat and drink, games, poetry, and sex.

It's not hard to imagine the Sapphic circle as a kind of feminine counterpart to that. The poetry would be the "poetry" part; readiness for wedlock would be a valuable aspect of becoming a member of society; and the "sex" part would be one aspect of the attachments within the social group. Unfortunately, we have very little information about women's social groups in ancient Greece, so this is more a plausible reimagining of Sapphic society than anything more definite. But the surviving bits of Sapphic poetry sure look like the kind of thing we'd expect to see in that kind of context.

André Lardinois, whom Mendelsohn mentions in the article, is the best person to read on this subject. I assume Mendelsohn is referring to Lardinois's introduction (which I haven't read) in Diane Rayor's recent translation of the fragments. If you want to delve into Lardinois's more academic writing, there's a famous article by him called "Subject and circumstance in Sappho's poetry" in Transactions of the American Philological Association 174 (1994): there he goes over several representations of Sappho as "school-teacher" -

Holt Parker's reconstruction of Sappho as a symposiast and... the other modern images of Sappho: teacher at a school, leader of a thiasos, and instructor of a chorus.

If you really wanted, you could make the word "school" fit any of those - it has a lot of different meanings depending on context. But it does run risks of people taking it too literally. So it's probably best not to talk about it as a "school".