Threat - if the program/project genuinely requires that the manager needs to on emails regularly and if you don't give what it takes to a manager of those programs/projects, they might replace you with someone who does.
Opportunity - if you check your emails in vacation and take proper action, then you will gain trust from management and they don't even bother whether you are in office or in vacation. they might even give more responsibilities.
Because business never goes on vacation, you cannot tell customers that key person is in vacation. You can find someone to act as a manager when you are on vacation. But that person might replace you, if you are not very careful.