They do. It is not normal, again, that organizations, public or private, would require proof of a health status or vaccination status. There have been exceptions. School children, for instance, need to provide proof of vaccination in order to enrol in school. In travel situations, there have been exceptions as well, but the rule is that this is confidential health information that should not be required except for very precise purposes.
On June 21st, 2021. See this statement in context.