This is a fascinating question, but quite a difficult one to answer for several reasons. The most significant issue with answering your question is with defining 'transgender.' It's a contested term in the modern world with plenty of serious academic debate, let alone the passionate disagreements in popular arenas. These modern debates are important, but I don't think it's something to get into here, so I propose to use the dictionary definition from here:
noting or relating to a person whose gender identity does not correspond to that person’s biological sex assigned at birth.
This is obviously reductive, but it may serve our purposes for now. The big question is whether we can apply this modern definition to the ancient world, where ideas about gender and sexuality were completely different to our own. I suspect that we probably can't, but before we get to grappling with that thorny issue, it might be an idea to look at the evidence for Elagabalus to see what we're dealing with. Once all our cards are on the table, so to speak, we might have a better sense of how to proceed.
I'll include here a short biography of Elagabalus for those unfamiliar with him (birth name Varius Avitus Bassianus, a.ka. Heliogabalus, Sardanapalus and many others. I'll stick to Elagabalus for clarity). He became emperor in AD 218, aged 15, as a figurehead of a rebellion against emperor Macrinus that was organised by his mother and grandmother (both relatives of Septimius Severus, emperor from 193-211). He was high priest of Elagabal (the god has as many names as the emperor, but I'll stick to Elagabal for clarity), a Syrian (sun?) god worshipped in Emesa (now Homs). On becoming emperor Elagabalus moved himself and the black stone representing his god to Rome, where his outlandish behaviour and various insults to traditional Roman morals quickly alienated him from the aristocracy and army. He was forced to adopt his cousin, Severus Alexander, only 4 years younger than him. Eventually the constant outrages towards the power-brokers in Rome grew too unacceptable, and Elagabalus was murdered in AD 222, having been emperor for only 4 years. The young Alexander took over, controlled by his mother and a council of prominent senators including Cassius Dio and the jurist Ulpian.
There are three main sources for the short reign of Elagabalus: Herodian, Cassius Dio and the Life of Elagabalus from the Historia Augusta. It should be emphasised from the outset that all three are incredibly hostile to the emperor, and therefore their accounts cannot be taken as objective fact. The HA is unreliable for a number of other reasons, and contains the most sensational and obviously fictional stories, so I'm going to primarily use Dio and Herodian here. Both authors were contemporaries, and at least Dio was in Rome at the same time as Elagabalus, so has some value as a potential eye witness. Nevertheless, and I stress this again because it's really important, both authors start from a position of open hostility towards the emperor. Their reasons for this hostility are primarily based on his religious practices, and that is certainly the focus of both narratives. Both Dio and Herodian use well-worn tropes about eastern luxury and decadence in their narratives, familiar to Greek and Roman literature for centuries by this point. For Dio this is also the crescendo of his whole work - a history that started with the founding of Rome in the mists of the mythical past and came right up to his own time. By describing Elagabalus in the way he does, Dio is trying to impress his audience with a big finish, and no doubt making moralising points about the state of contemporary Rome compared with its legendary past. Again, it's really important to take this all into account, and not to look in isolation at the passages I'm about to quote. Ancient historical writing is a genre of literature, written for an audience and with the intention to entertain as well as inform. This is not the dispassionate reporting that historians from other periods might be familiar with.
OK, so let's finally get to the evidence itself. There are several passages from both Herodian and Dio that are usually taken as evidence that Elagabalus was transgender. First, and most numerous, are descriptions of his clothing and behaviour:
Herodian
[5.5.4] His dress showed the influence of the sacred robe of the Phoenicians and the luxurious garb of the Medes. He loathed Greek and Roman garments because they were made of wool, in his opinion an inferior material; only the Serian cloth [i.e. silk] met with his approval. Accompanied by flutes and drums, he went about performing, as it appeared, orgiastic service to his god.
[5.5.5] When she saw what Elagabalus was doing, Maesa [his aunt] was greatly disturbed and tried again and again to persuade the youth to wear Roman dress when he entered the city to visit the Senate. She was afraid that his appearance, obviously foreign and wholly barbaric, would offend those who saw him; they were not used to such garb and considered his ornaments suitable only for women.
[5.6.10] He had no desire to sin in secret, but appeared in public with eyes painted and cheeks rouged; these cosmetics marred a face naturally handsome.
[5.8.1] With everything that formerly had been held sacred being done in a frenzy of arrogance and madness, all the Romans, especially the praetorians, were angered and disgusted. They were annoyed when they saw the emperor, his face painted more elaborately than that of any modest woman, dancing in luxurious robes and effeminately adorned with gold necklaces.
Dio
[80.14.3] When trying someone in court he really had more or less the appearance of a man, but everywhere else he showed affectations in his actions and in the quality of his voice. For instance, he used to dance, not only in the orchestra, but also, in a way, even while walking, performing sacrifices, receiving salutations, or delivering a speech. 4 And finally, — to go back now to the story which I began, — he was bestowed in marriage and was termed wife, mistress, and queen. He worked with wool, sometimes wore a hair-net, and painted his eyes, daubing them with white lead and alkanet. Once, indeed, he shaved his chin and held a festival to mark the event; but after that he had the hairs plucked out, so as to look more like a woman.
[80.16.3-4] Elagabalus, on seeing him, sprang up with rhythmic movements, and then, when Aurelius addressed him with the usual salutation, "My Lord Emperor, Hail!" he bent his neck so as to assume a ravishing feminine pose, and turning his eyes upon him with a melting gaze, answered without any hesitation: "Call me not Lord, for I am a Lady."
Next we have descriptions of his sexual activities:
Herodian
[5.6.2] So that he might seem to be doing something manly**, he made love to one of the Vestal Virgins of Rome [Julia Aquilia Severa], priestesses who are bound by sacred vows to be chaste and remain virgin to the end of their lives; taking the maiden away from Vesta and the holy virgins' quarters, he made her his wife. He sent a letter to the Senate asking to be forgiven his impious and adolescent transgression, telling them that he was afflicted with a masculine failing - an overwhelming passion for the maide**n.
Dio
[80.13.1-3] He married many women, and had intercourse with even more without any legal sanction; yet it was not that he had any need of them himself, but simply that he wanted to imitate their actions when he should lie with his lovers and wanted to get accomplices in his wantonness by associating with them indiscriminately. He used his body both for doing and allowing many strange things, which no one could endure to tell or hear of; but his most conspicuous acts, which it would be impossible to conceal, were the following**. He would go to the taverns by night, wearing a wig, and there ply the trade of a female huckster. He frequented the notorious brothels, drove out the prostitutes, and played the prostitute himself. Finally, he set aside a room in the palace and there committed his indecencies, always standing nude at the door of the room, as the harlots do, and shaking the curtain which hung from gold rings, while in a soft and melting voice he solicited the passers-by