This discussion is relevant to your interests.

So the first thing to keep in mind is that "anti-vax" is not a monolithic camp. It's a constellation of complimentary suspicions and tribal allegiances. The second thing to keep in mind is that the tribal allegiances matter far more than the suspicions. By keeping their kids unvaccinated they're not succumbing to fear, they're practicing an alignment ritual with their chosen tribe. Fortunately this tribe is loosely coordinated and its rituals and rites are ad-hoc.

All you really need to do is winnow the "I do not vaccinate my kids" thread away from the "I am a left-voting liberal that eats organic, supports Heifer International, enjoys kombucha and occasionally reads Goop" threads. This may or may not be possible, depending on their identity and how invested they are in it.

The principle advantage you have is that most anti-vax sentiments are lightly examined and poorly understood, and often the act of requesting someone explain their understanding of the issue to you forces them to organize their thoughts, examine the facts they hold and synthesize a position. As a biologist you're in a great standpoint to honestly question what sort of information they're seeing and why they judge it credible.

The trick is not to say "show my why you believe this so I can tell you you're full of shit" it's to say "I haven't seen anything compelling that makes me doubt the efficacy of vaccines; I'm curious what you're reading."

If you can safely and comfortably show her that you aren't rejecting her tribe, but that there might be some issues with certain well-proscribed areas of dogma, you can get her to question that dogma without excessive cognitive dissonance. If on the other hand she's like a Jehovah's Witness you're fucked.