You really should talk with a priest. Issues around cooperation with evil are pretty complicated. You need to figure out if it is material or formal cooperation. Personally, I would say it is material cooperation which, along with other factors, leads me to the conclusion that you could give out the vaccine forms. But I don't have the level of expertise you need.
There is a Vatican document on the issue, if that is a source you could trust:
https://www.immunize.org/talking-about-vaccines/vaticandocument.htmThis document does a decent job of explaining the traditional principles of moral theology involved, so it would be a starting point, at least.
The principle of licit cooperation in evil
The first fundamental distinction to be made is that between formal and material cooperation. Formal cooperation is carried out when the moral agent cooperates with the immoral action of another person, sharing in the latter's evil intention. On the other hand, when a moral agent cooperates with the immoral action of another person, without sharing his/her evil intention, it is a case of material cooperation.
Material cooperation can be further divided into categories of immediate (direct) and mediate (indirect), depending on whether the cooperation is in the execution of the sinful action per se, or whether the agent acts by fulfilling the conditions - either by providing instruments or products - which make it possible to commit the immoral act. Furthermore, forms of proximate cooperation and remote cooperation can be distinguished, in relation to the "distance" (be it in terms of temporal space or material connection) between the act of cooperation and the sinful act committed by someone else. Immediate material cooperation is always proximate, while mediate material cooperation can be either proximate or remote.
Formal cooperation is always morally illicit because it represents a form of direct and intentional participation in the sinful action of another person.10 Material cooperation can sometimes be illicit (depending on the conditions of the "double effect" or "indirect voluntary" action), but when immediate material cooperation concerns grave attacks on human life, it is always to be considered illicit, given the precious nature of the value in question11.
A further distinction made in classical morality is that between active (or positive) cooperation in evil and passive (or negative) cooperation in evil, the former referring to the performance of an act of cooperation in a sinful action that is carried out by another person, while the latter refers to the omission of an act of denunciation or impediment of a sinful action carried out by another person, insomuch as there was a moral duty to do that which was omitted12.
Passive cooperation can also be formal or material, immediate or mediate, proximate or remote. Obviously, every type of formal passive cooperation is to be considered illicit, but even passive material cooperation should generally be avoided, although it is admitted (by many authors) that there is not a rigorous obligation to avoid it in a case in which it would be greatly difficult to do so.
