There were, of course, always plenty of rumors about the schemes and misdeeds of various emperors. In some periods, likewise, aristocratic Romans suspected that the Empire was really being run by imperial freedmen - see, for example, Seneca's very strange Apocolocyntosis divi Claudii ("Pumpkinification of the deified Claudius"), which shows that emperor as (among other things) a tool of his former slaves.

In Late Antiquity, there were also dark rumors about the sympathies of some Germanic court officials - the Gothic generalissimo Gainas, for example, was suspected of being in league with Rome's barbarian enemies.

The closest equivalent of a conspiracy theory I could think of, however, was the Roman government's panic about Manicheism at the end of the third century. Manicheism was a dualistic religion established in the mid-third century by the prophet Mani, who lived and taught in the Mesopotamian heartland of the Persian Empire. Mani's disciples quickly spread the new religion into the Roman Empire, inducing something of a panic in certain quarters. We don't know how widespread distrust of the Manicheans was, but the emperor Diocletian and his officials regarded the new religion as nothing less than a Persian-engineered conspiracy designed to overthrow the Roman Empire. We have a remarkable letter from Diocletian and his co-emperors to the proconsul of Africa (modern Tunisia) that reads in part:

" As regards the Manicheans....who, in opposition to the older creeds, set up new and unheard-of sects, plotting in their wickedness, to cast out the doctrines granted to us by Divine favor in ancient times, we have heard that they have but recently emerged or sprung forth, like strange and monstrous portents, from their native homes among the Persians — a nation hostile to us — and have settled in this part of the world, where they are perpetrating many evil deeds, disturbing the tranquility of the peoples and causing the gravest injuries to the communities; and there is danger that, in process of time, they will endeavor, as is their usual practice, to infect the innocent, orderly and tranquil Roman people, as well as the whole of our Empire, with the damnable customs and perverse laws of the Persians as with the poison of a malignant serpent." (Mosaicarum et romanarum legum collatio, 15.3.3-4)

Just to prove they aren't messing around, Diocletian & co. order the governor to burn the leaders of Manicheans alive atop piles of their scriptures (loc. cit., 15.3.6).

There were some conspiracy theories attached to Christianity when it first started to be noticed by the Romans. For example there were rumors that Christians literally ate people, which likely stemmed from the Eucharist and the whole "body and blood of Christ" thing.

Here is a link to the internet history source book that shows some of the accusations against Christians: https://sourcebooks.fordham.edu/ancient/christian-cannibals.asp

They know one another by secret marks and insignia, and they love one another almost before they know one another; everywhere also there is mingled among them a certain religion of lust, and they call one another promiscuously brothers and sisters, that even a not unusual debauchery may by the intervention of that sacred name become incestuous

Note the idea of "secret marks" and insignia, which aligns closely with a lot of modern conspiracy theories. I also find it interesting that this guy is offended by the language of "brothers and sisters" and the fact that Christians talked about love in a way he views as almost promiscuous. I would love if someone with more knowledge of Roman culture would expand on this.

And the classic "they eat babies" conspiracy:

Now the story about the initiation of young novices is as much to be detested as it is well known. An infant covered over with meal, that it may deceive the unwary, is placed before him who is to be stained with their rites: this infant is slain by the young pupil, who has been urged on as if to harmless blows on the surface of the meal, with dark and secret wounds. Thirstily - O horror! they lick up its blood; eagerly they divide its limbs. By this victim they are pledged together; with this consciousness of wickedness they are covenanted to mutual silence.

The "they eat babies" conspiracy is kind of "old school", you don't see it around AS much anymore, but it used to be an accusation thrown against Jews in antisemitic conspiracies, as well as "they steal babies".

Follow up question: how well are we able to answer the OPs question? I am not a historian, but a chemist so forgive my ignorance. My understanding is that much of the written sources we have from the distant past come from the writings of 'nobles', clergy and other high ranking people as they were actually literate (and I assume their writings are more likely to have been cared for - correct me if I'm wrong). Is it not possible, perhaps even probable, that we could be missing widespread conspiracy theories? Obviously we know of some from the other posts, but I'm just curious how a proper historian would even try to approach this question.

My apologies if methodological questions are not allowed.