I actually do not think they should be able to do that for school children either. Yes it is similar yet I also see it as authoritarian.

If the school is a voluntary choice then I wouldn't have a problem with it. As soon as they start dictating when, where, and how a parent is permitted to school a child then it should not be permissable.

If people were free to send their children to schools that did not have such requirements, or free to send them to ones that do so it becomes a choice then I would not have a problem with it.

A lot of the times they'll say "you can homeschool" them which they try to discourage as well but often due to how society is structured that is unrealistic for many and thus becomes a kind of coercive force. EDIT: Some states mandate vaccines for children even if they are homeschooled.

It is also important to differentiate between a business and a government funded entity that is funded by taxpayer taxes which are not exactly something you can voluntarily choose to provide or not provide.