By design in Front, when you hook up an email channel to your individual inbox, no other teammate will be able to send from that address-- even if you delegate the inbox. 

(More on individual inboxes here, and more on delegating here).

However, there are a couple of options for being able to send from another user's email address.

The first option is to create a shared inbox and connect the email there. You can toggle the settings for this inbox so that only certain teammates within the company have access. It's important to note that those teammates will be able to view all of the conversations within that inbox, which might not be what you're looking for. 

Another option would be to keep the email address connected to a private inbox, but to use the Share Draft functionality. In this workflow, the owner of the private inbox can share select conversations with a teammate, who can jump in and compose an email on behalf of the inbox owner. However, the actual inbox owner will need to hit send for them.