I believe that some vaccines don't use "aborted DNA". I recall this website has some lists:
https://cogforlife.orgAs for handing out a vaccine form, I don't know. Does the form mention the name of the vaccine? If so, maybe check the above website and see if you see it on one of the lists, and, if it does use "aborted DNA", I'm inclined to think that you must refuse to give out the forms. But if you don't know whether or not the vaccine in question uses "aborted DNA" and/or is otherwise evil, then I don't know what to do.
This is why I personally cannot find employment. It shouldn't even be an issue, but it is. Every employer is evil, and there is no escaping the evil. Even the simplest tasks are made hopelessly complicated, because the employer is evil. You can't even work a seemingly-innocent job without being told to do something that's at least questionable.
So I don't know.
But I would avoid listening to any philosopher who tries to distinguish between "formal" and "material" cooperation, and any ethicist who tries to appeal to the "double effect". I don't trust them or their teachings. I'm pretty sure that ethics is the sort of science that can only be known by way of revelation. Human reasoning, in principle, is insufficient. You cannot reason your way to moral truth. Unless these principles are to be found in official Church teaching, I don't trust them.
And why must we all become expert theologians in order to work at the local post office, supermarket, or school? It's madness!