[ceothrowaway]

There is both opportunity and threat hidden here.

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.

[contravariant]

Both of those seem to imply that checking your email is the right thing to do. So just for balance let's consider the following as well:

Threat - Checking your emails means agreeing that it is normal and to be expected, this will be hard to undo.

Opportunity - Not checking email for a few days will force both yourself and the team to be more independent, and will identify obstacles to further independence.