This started in the UK in the 90s with a research paper by Andrew Wakefield linking MMR (measle mumps rubella) vaccines to autism. It was trash research but it for published in the Lancet (a major international journal) before being retracted once other doctors pointed out the massive flaws in the research.
There was and is no evidence of causation - autism happens to be diagnosed usually after childhood vaccines are conpleted but thats because vaccines are mostly in young ages and it takes a while for autism to be diagnosable as its only obvious once children reach a certain age when the socialization aspects of the diseases become more obviously.
However despite it being trash research and eventually being withdrawn, the damage was done. Enough parents of children with autism wanted to believe that this disease was inflicted upon them and have someone to blame rather than accept it is largely genetic and bad luck. A perhaps understandable feeling but that gave an opening for conspiracy theorists to blame the government for a "cover up" even though all the counter evidence and push pack is evidence based and freely unavailable.
Andrew Wakefield eventually got struck off the UK medical register - he was found to have had undisclosed financial interests that would make him millions in selling bogus test kits. The real conspiracy was his but Hes managed to move to the US and make a career as a "victim" and "outsider" to the pharmacy industry.
This whole vaccine conspiracy has been taken up with the US right wing and religious groups. Its a perfect conspiracy for them as it plays into the ideas of the US federal "forcing" then to do things against their will. In this case vaccinating children (which depends on a majority of children getting vaccinated to protect the whole population - herd immunity) and is used as an example of "socialism" vs their preferred extreme individualism. They already rail against being told they cannot indoctrinate children by lying about science in schools (trying to suppress evolution teaching etc) or use the states infrastructure to discriminate against groups they disagree with such as gay or trans people, or be downright racist asis often seen throughout the bible belt.
So the vaccine conspiracy theory is basically one of many tools used by the right wing and religious allies to rail against supposed state interference in their lives. Instead most people who believe in this nonsense are either extremely ignorant and easily manipulated or deliberately using the nonsense to further their own goals. So some of these people are highly intelligent and don't care whether this is true or false - only that it aligns with their world view and goals so they dont challenge it. Some will even know its all bullshit and go along with it to further their own goals.
The covid vaccines has supercharged this debate. The roll out of vaccines with massively reduced testing and safety steps to try and control the pandemic, and then the side effects seen has all helped fuel this conspiracy and grow it within the right wing echo chamber.
There is no evidence whatsoever that vaccination causes autism. However parents are refusing to have their children vaccinated with MMR and now you have outbreaks of diseases like Measles in the US. People will die, people will become infetile - all from a disease that is easily prevented by a vaccine.
Tl:dr: The vaccine conspiracy is a right wing aligned nonsense started in the 90s andnuper charged by covid, and is a sign of the extremely polarised and disinformation heavy nature of right wing US politics (and is seen in other western countries if you dig into it even if fringe stuff)
