Connecticut Gov. Ned Lamont said on Monday that the state should join a handful of others that require minors to receive vaccinations for preventable diseases, eliminating most exceptions to vaccination.

“The more children who receive their vaccinations, the safer it is for everyone, especially those who may be at risk to catch serious diseases,” Mr. Lamont said in a statement. But if a parent decides not to vaccinate “then alternate decisions must be made about where to educate your children,” he said.

During...