I think a lot of ammo for anti-vaxers comes from the government and media not being completely honest about vaccine side-effects. That's because it is difficult for an individualist or a socialist to separate public health from individual health. The CDC is not your personal physician. The CDC worries when newspapers publish facts about vaccines, such that some vaccines can cause severe and traumatic side effects, while being 52% effective, because they worry about obtaining herd immunity and need people to accept the vaccine for that. They want you to go through weird little side-effects, like not enjoying alcohol for months, so you don't infect a senior they count in the flu statistics. It's when "do no harm" means sacrificing the lives of a few to save the many.
Then there are political and funding concerns. If your own government funds your research to investigate if Agent Orange is harmful to the point of culpability costing billions, you are very careful and delicate with your conclusions. If your vaccination tests show that black Africans show more severe adverse effects, or that, maybe yes, the connection between vaccination and (worsening of) autism is both plausible and understudied, what do you do? For them its a single kid, maybe 2-3 in 100.000 which will start to suffer from autism years earlier, and a disease eradicated. For anti-vaxers, it is their kid.
Essential workers and at-risk people are first in line to receive the vaccine. The hope was availability in September, and public rollout in the beginning of 2021.
The anti-vax conspiracy mud has completely destroyed online information about this subject. There are more such subjects, where it is very difficult to find what the anti-activists are freaking out about. For instance, my gut tells me that physical and mental torture can be effective to obtain useful information. That that's why the mob, military, and police mainly use it for. But, online, nothing: torture is completely ineffective for the first 10 pages of Google. The Holocaust is another obvious one. 5G rollout ("completely safe, maybe, we don't know for sure. Improved Youtube streaming is important tho!"). Or try to find information on how the riots created a spike in infections. All news sites report on the same non-peer reviewed non-scientific institute paper, saying the riots caused spikes is a lie, yet the paper clearly postulates that riots increased infection for black participants and their families, but taken on the whole, they likely caused non-protestors to stay at home more for fear of violence and this reduced virus spread.