The carnist claims that vaccines (or any medicine) contain animal products or are tested on animals and therefore are not vegan, and vegans that consume such products are therefore hypocritical.
If anything, such arguments should be interpreted as a call for more veganism. That is, we need to create a quorum to push animal cruelty out of the market of medicine altogether. But there's zero leverage or pressure there so long as animals are consumed in meals. I mean, why spend research money to eliminate a rounding error on the fraction of animals abused annually? There's no way to push that agenda until we're on board about food.
Veganism is engaged in harm reduction where possible. If it's not possible to reduce harm beyond choosing humans (yourself and those around you) and the animals.... then there's no choice to make that eliminates harm. So this is no better an argument against veganism than suddenly allowing human murder because of the first instance of killing in self-defense ever.
The principal reasons to be against carnism is that it causes harm to conscious beings that don't deserve it, it causes ecological damage that hurts other people, and it hurts yourself. Vaccines barely satisfy one out of the three, (and only then if you take a hard deontological view): they, in general, help reduce infection amongst - for instance - immunocompromised or elderly people and reduce the harm to them from disease. So they are, in the aggregate, a means of suffering reduction. They also don't have significant volumes to make a substantial ecological impact (one jab a year from a fraction of an egg vs eating a whole animal each meal), and they protect you against disease. So, even while containing animal products, it satisfies none (or kind of, barely, but not really, one) of the arguments that vegans make against carnism. So there really is no comparing getting a vaccine to eating animals at each meal.
Just because someone else does something unethical does not justify your unethical action. Pointing to vegans as hypocritical for engaging in some behavior doesn't actually demonstrate a coherent argument against veganism.
If you want to state it as "why go vegan when it's impossible if you still want to be vaccinated", then great, go vegan except for vaccines. You're reducing your impact by orders of magnitude, which is always good. And maybe you can't eliminate it, but you don't need to eliminate all crime everywhere to not murder someone.