I’m of two minds:

People had a knee jerk reaction to the concept of a vaccine requirement, even though they largely weren’t a thing, and didn’t want the government telling them what to do. From there, if the government telling you to take it is bad, then it must be bad, because why would the “bad guy” try to push something good? That made people focus on the real negatives out of context and proportion to reality. It is newer technology that doesn’t have as much testing as other vaccines. That ignores that there’s no theoretical reason for it to be different, and that it’s less tested compared to some of the most tested things ever, and still actually very tested.
Then they started seeing the very complicated approval process that looked like it was being skipped, rather than skipping paperwork and running multiple safe trials at once when they’re justified instead of one at a time.
Combine that with how any major event causes people to become conspiratorial and you end up with vaccines being a “thing”.

Alternatively, if people wanted to intentionally fuck over the country because they were Russian assets, they might do stuff like fire everyone in a bunch of government agencies, maximize chaos to reduce their effectiveness, do random things to tank our stock market and slow the economy, spread disinformation about vaccines and try to get people to avoid them, and then suppress information about the spread of infectious diseases to get a lot of people sick and dead, and generally throw the country into chaos.

The first one looks exactly like what we all saw happen. There’s no evidence for the second one except that if we take motivation out of the picture we wouldn’t be able to tell what’s happening apart from the second scenario.