The "trolley problem" is a philosophical dilemma regarding the morality of allowing harm caused by inaction versus causing harm by direct action. In a common form of the dilemma, you are faced with a situation where a train trolley is headed to hit a number of people. You have the option to pull a lever that will direct the trolley to hit a smaller number of people, thus actively condemning that smaller number.
The argument can be made (though I disagree with it), that you are "more responsible" for deaths caused by pulling the lever than those caused by inaction. Whether or not you agree with this argument, plenty of research has shown that humans are typically very averse to causing known harm by direct action, even though they know that inaction will cause greater harm. That is: "Human instinct" in the trolley problem seems to be to not pull the lever, even though a majority of people agree that it is the morally correct decision.
Finally, the above discussion is more or less completely independent of the number of people on the two tracks. Whether it's 1:2, or 1:1 000 000 doesn't really change the moral argument of action vs. inaction. If you get into the kind of numbers where the number of deaths would have global secondary consequences, you have a new kind of dilemma though. However that's not the case here, hence my choice of pulling two arbitrary numbers out of a hat, rather than using actual numbers for vaccine side effects and infection lethality.
What I'm doing here is noting that the question of vaccinating or not is philosophically equivalent to the trolley problem. Therefore, we can use research of the trolley problem to understand how vaccine sceptics think, and to learn something about how people can be convinced that pulling the lever is the morally correct choice.