OP you should publicly name the agency.
Unnecessary. If someone is concerned and makes a point of asking whenever they are booking and the agency declines to answer, then you politely say goodbye. The end result is the same if you know in advance or not.
In fact it is better not to know. That way if the agency gets a bunch of calls asking about vax status and don't answer and all those callers say that they won't book because of that, the agency may change their minds about disclosing because they are losing business.
