Thanks for suggestion.

WHMCS will automatically filter emails if a number of auto-responders are received in a 15 minute period: https://docs.whmcs.com/Support_Tab#Ticket_Reply_Email_Limit
This setting could be reduced to stop more than 1 or 2 auto-responses. The difficulty in preventing all emails entirely is being able to reliably determine an auto-responder from a genuine message, there isn't anything in the email headers themselves to differentiate them.

If there were any suggestions on how this might be achieved, that would be a good topic for discussion.