“Have you had your vaccine?” is not an appropriate question to be asking of others, publicly or especially in a healthcare setting. The health-status of an individual is a private matter and people may have many reasons or hidden conditions they have no wish to reveal or discuss without proper justification.
A practitioner may need to ask about a client's vaccine status, to understand, for example, if they could be suffering from adverse reactions. But this is not an appropriate subject of social conversation if there is no specific medical need to know.
It is the Oxford Street Therapy Centre's policy that vaccine status, of staff and clients, should not be enquired of, discussed or questioned without good medical justification being clearly stated in advance.