Is it permissible for the muezzin to go for 'Umrah and delegate someone else for the Adhan?

It is not permissible to abandon your work and assign it to someone else without the permission of the responsible authority. This is a trust and a duty that has been entrusted to you, and for which you receive your wage. Allah (the Exalted) says: "Indeed, We offered the Trust to the heavens and the earth and the mountains, but they declined to bear it and feared it; but man [undertook to] bear it. Indeed, he was unjust and ignorant." It is permissible to be absent if it is customary or with the permission of the responsible person.