I think there might be two things that make it hard to answer the question. The first is that "always" is a remarkably long time, and people can probably only speak to specific time periods. And the second issue is what exactly counts as a "conspiracy theory", especially in pre-modern times, but even in modern times, can be somewhat controversial.

To restrict this a bit and speak of conspiracy theories in the modern US, I would say that obviously not all conspiracy theories are conservative. There are conspiracies for example that AIDS was introduced to the black population by the US government. Or there is the Dark Alliance claim, where the government purposefully flooded crack cocaine into black neighborhoods. Although that one that leads us to the second issue. Because there's some debate about how much of that is a "conspiracy theory" with authors like Cockburn in WhiteOut [published by Verso so take that for what it is] arguing that the facts actually do support much of the Dark Alliance theory and that the parts that are unsupported are exaggerations that were not found in the original argument. If we limit ourselves just to conservatives, there were plenty of fascists who found something in the Protocols of the Elders of Zion.

So I think that we can at least say that there have been non-conservative conspiracy theories.

Michael Barkun might go a bit further though. In his A Culture of Conspiracy, he argues that we have recently witnessed a major change in how conspiracies work in that right-wing conspiracies seem to have merged with nonpolitical or even left-wing stigmatized beliefs.

Once there were different conspiratorial communities: your New Agers, your right-wing antigovernment extremists, your Ufologists, your flat-eathers, etc. (you might not accept some of these groups as conspiracy theory groups, but I think one would accept at least some of them as conspiracy theorists or, at the very least, as promoters of stigmatized knowledge). Initially these communities were rather distant from each other, but in the 80s and 90s crossover began to happen. You would have David Icke presenting at both militia groups and at New Age groups. The New World Order conspiracies started to leak from Christian fundamentalists and right-wing militias into the UFO subculture. At the same time, the media shifted more towards valorizing conspiracy theories (think of the X-Files; there's a conspiracy involving the government and aliens, you're told to trust no one, etc.), creating an opening for new people into the new conspiracy theory milieu.

So, if Barkun is right, then conspiracy theories -- or at least communities dedicated to different forms of stigmatized knowledge -- were not necessarily or always associated with conservative politics. Instead, conspiracy theories had conservatives and non-conservatives who used to keep to their own groups in their own conspiracies. But sometime in the 80s the boundaries between the two began to collapse, certain fair-right conspiracy theories ended up combining with more neutral conspiracy theories (e.g. NWO + UFO), and the different milieus began to merge. On the one hand meant that some of the more right-wing theories lost some of their racist or anti-Semitic aspects (e.g. Icke might use the Protocols of the Elders of Zion, but will claim that his Reptilians have nothing to do with Jews), but on the other hand meant that there was a ready made bridge between the more less avowedly right-wing theories and the far-right theories (e.g. There are other groups who do use the Protocols and believe they are for Jews and if you are used to trusting the Protocols and are in the conspiracy theory milieu, then you might be more willing to trust that it actually does mean Jew).