General Note
People often have different ideas about what “mandatory” means, so let’s draw a distinction between conditional and unconditional mandates. A conditional mandate is when individuals are allowed to opt-out, but there is some sort of cost for doing so. An unconditional mandate does not allow people to opt-out and includes measures to force compliance.
So while school attendance is mandatory in the US, no one goes out and drags absent students to class. Parents face fines and even jail time for not sending their kids to school, but the students are never forced to sit down in front of a teacher. And of course, there are options for fulfilling the mandate other than the public school system (such as private schools and homeschooling).
The existence of a mandate, then, doesn’t mean anyone can be forced to comply. Only an unconditional mandate goes that far.
Specific Answer
I think we should have a carrot and stick approach that still preserves people’s individual freedoms (in short, a conditional mandate). The carrot? All basic vaccines are provided free of charge to the recipient provided they are received on schedule. Alternatively, children with anti-vaccine parents can receive them for free when they achieve the age of majority and no longer need the permission of a parent.
The stick? If they aren’t willing to participate in public safety measures, they don’t have to be allowed into (certain) public spaces. So unvaccinated children can be prohibited from attending public schools, for example, and their parents must abide by all laws concerning the education of children (meaning that they must either homeschool their children or send them to a private school that accepts unvaccinated children).
Exceptions should of course be made for children who have valid medical reasons for not receiving the vaccines.