Vaccines save lives. I'm not here to argue about that, take it somewhere else. If you don't agree, for the purposes of this question just imagine I'm talking about some medicine that accomplishes vaccination, but is magically safe from whatever invented danger that bothers you about it.
Vaccines save lives. However, they are not administered to treat a current illness -- they are given to prevent dangerous illnesses from coming.
Since we know that one may only violate Shabbos for a חולה שיש בו סכנה (patient who is in danger for his life) for things that have to do with saving that patient's life (see Rambam, Hilchot Shabbos 2 (English), Shulchan Aruch, Orach Chaim 328), if we would have to be מחלל שבת in order to administer vaccines1, would we be allowed to violate Shabbos to administer vaccines?
1 It's debatable whether any injections are Biblical Shabbos transgressions, see שמירת שבת כהלכתה ch 32:58, and footnote 151 there; it appears that all agree that subcutaneous and intramuscular injections are not biblically prohibited. For the purposes of this discussion, let's assume that there is some universally agreed-upon Biblical transgression involved, such as driving a car in order to get it, or writing a prescription for it.