The most famous name is probably Dora Richter. She worked directly with Hirschfeld and she's usually regarded as the first known person to receive a vaginoplasty (in 1931). She had a very difficult life and disappeared off the face of the Earth in 1939 though.

Unfortunately though history wasn't great at recording the names of the various queer people who were clients at the institute. For one, most of them wanted privacy. 1930s Germany wasn't exactly the greatest time to be a publicly known queer person, so many of the records used pseudonyms or abbreviations of names. A lot of the recorded names are just a first name, last initial, and address, for instance. Second, the Nazis burned the Institut's archives, so much of the notes/research are lost.

I should also point out that Hirschfeld himself was a gay man. His partner, Karl Giese, was the Institut's head archivist and he was killed by the Nazis at the Lodz ghetto in 1942. Any amount of research you do into the history of this place will probably fill you with terrible sadness and I'm sorry. They were really ahead of their time, reaching near modern levels of understanding of queer people, and yet the entire thing was stomped to death by fascist scum.

Like axont said, the unfortunate reality is that the vast majority of these people are anonymous to the historical record. If you're interested in this history then I highly recommend Gay Berlin: Birthplace of a Modern Identity by Robert Beachy and The Hirschfield Archives: Violence, Death, and Modern Queer Culture by Heike Bauer, both of which are on libgen. The latter has a little bit about Dora Richter in chapter 4 under the heading A Space for Transgender, pages 86-87 in the hardcopy (not sure if the libgen version aligns with that or not). Another very interesting, albeit short, source on this kind of self-advocacy can be found in the account of an American doctor, William J. Robinson, who visited the Institute in 1925: https://hdl.handle.net/2027/mdp.39015011423632?urlappend=%3Bseq=397%3Bownerid=13510798887791650-401 (pages 391-396 if that didn't link properly). He inaccurately uses the term 'homosexual' for every queer person he talks about, but it's pretty easy to pick up on who's trans from what he descibes. His description of Hirschfeld and the Institute is absolutely glowing, he seemingly can't heap enough praise on the work they were doing. One of my favorite parts is how he laments that he would love to set up a similar institution in the US, but deems American society to be far too reactionary to allow it,

"The scope of the Institute is a much wider one, embracing as it does the entire field of sexology. It is an institution absolutely unique in the whole world. It is an institution of which I dreamed for many years and which I hoped to establish in the United States but which I felt would not thrive on account of our prudish, hypocritical attitude to all questions of sex. In such an institution one has to have a free hand; the advice given must be unhampered by any fear of violating some medieval law or of colliding with a stupidly childish, and for that reason all the more tyrannical public opinion. What can be done in barbarous Europe cannot always be done in ultra-civilized North America. [...] The United States could certainly use five or six such Institutes—say one in New York, one in Boston, one in Chicago, one in Atlanta, one in San Francisco. They would all have plenty of work to do, and less ignorance and consequently less misery in sex matters would be the result. When I get back to New York I may try to establish the first Institute of this kind in the United States—thus doing on a large scale what I have been doing on a small scale for a quarter of a century. And then again I may not: too much hard, nerve-wrecking work."

Anyway, a bit more on topic: his article ends with a very bittersweet sentiment that I think about all the time, "May the [Institute] be built on a permanent, never to be shaken foundation." Fuck this billionaire freak for denying this horrific tragedy that saw the loss of invaluable queer history and research, and set us back decades in medical development and understanding of queer people.