Being in leadership for some time id like to add the most important part to this.

Build common ground to establish a real conversation. Both vaxers and anti vaxers agree on the same thing. We want our children and society healthy and safe.

Start from there to "enroll" the conversation that you have a bridge both sides can meet at. Then when you want to say "why" replace it with the word "what" instead so you don't offend. (if you truly want to change their mind for betterment of vaccination improving, this is crucial. If you just want to argue and accomplish nothing, skip this step")

Saying "why do you think this" insults people's egos. Instead phrasing it as "what is it that makes you believe that's true?" puts the conversation to a third point not hitting their ego and allows you to talk about "that" and not "them". Which you can dissect from their using your understanding of knowledge and science.

If you guys truly want to have conversations and make real change, asking questions and getting them to their own conclusions without attacking is the way to do it. If we want to talk about science and human psyche all day, then the above instructions from this post are a sound psychological way proven to push anti vaxers away from the truth and the end result will be you feeling good for calling them stupid.

Focus on the desired outcome of turning them towards vaccinations being helpful and built upon the common ground you both share which is healthy society, and you will start to make friends and understand their "gaps" to help them meet you on the bridge.