This is a question governed by your bylaws and state law. Since the bylaws give the board the power to fill a vacancy by electing a person to serve out the balance of the term, apparently without providing any additional conditions, I assume you don’t have to get approval from any members (if you have members) and the board can take action on the election as it would on any other matter. Normally that would be done at a meeting, probably according to a bylaw provision that says action is taken by majority of those present and/or voting at a meeting at which a quorum is present.

Since it doesn’t appear that you intend to have an actual meeting, you can undoubtedly act by a unanimous written consent from all directors. Some states may allow a less than unanimous vote, but that is more likely to be the case with voting by members than with voting by directors.  If you are required to obtain unanimous consent, that means everybody, not 12 out of 13 or some other number less than all. As a practical matter, if you have overwhelming support but can’t get everybody to respond by email, you can invite the person to attend the next board meeting and take a vote before the person is seated. You don’t need unanimity to take action at a meeting.