Let’s first establish how we don’t help them change their minds.
Bombarding them with medical or scientific proofs or facts. Those have been around plentifully. Adding to them is not going to help.
Arguing. Arguing often makes people defensive and strengthen their positions. Especially if they assume that you’re hostile or fooled, and they are privy to crucial information you’re lacking.
Marshall Rosenberg (Non-Violent Communication) stresses: Empathy Before Education. You want to teach something; first, show empathy.
Many anti-vaxxers feel under attack. Some of them are. But even if their fears were paranoid, not substantiated by objective fact-finding, it’s worth it to hear and acknowledge how they feel (and sometimes are) victimized.
Agree where you truthfully can. “They’re lying!” Yes, unfortunately, there is a lot of lying going on on both sides of the debate. “Big Pharma just wants to make money.” Yes, the pharmaceutical industry is as much into ethics as the weapon and the tobacco industry. “This is just like the Holocaust.” Yes, you feel threatened in your survival and abandoned by all.
Notice that when you’re mild and friendly, their pronouncements will also soften. Don’t then say: “Ah, you admit it’s less bad.” Don’t destroy the new safety they’re building with you, just because you feel you need to win.
You could say: “People who feel so protected by G^d that they don’t need to look both ways when crossing the street also don’t need vaccinations,” but then, say it very calmly. Be very friendly when you say: “I wonder if mandatory vaccination is different from smoking bans and mandatory safety regulations like wearing safety belts or helmets.” Smile!
When you feel like screaming at them or crushing them with decisive contra-arguments, don’t. Don’t ignore this anger and shove it under the carpet (or it will burst forth eventually). Just promise you’ll say it later. Have some empathy for your battled self, praise your patience and love.
The most important thing is to keep them talking by being interested in what they have to say. This is not for a rush job.
When they say feeling deeply unseen, abandoned, etc., say casually: I guess that it’s not the first time in life you feel like that. Ask them when and how that started, and how it’s been ruining their lives. Show empathy for how bad it was. When they feel you’re on their side, say casually: “This injustice has to stop, and your hurts need to heal. As long as that has not happened, you’re vulnerable to people who want to manipulate you.”
“Good for you for seeing injustice and caring enough to oppose it. Yet, being angry makes you powerless. Try staying calm, happy, to gain power.”
When in doubt, be on their side instead of trying to get them on your side.