This is a question of state law and will be controlled by the state law default provisions if not covered in the organization’s bylaws.  The normal practice is to have a successor elected to fill a vacancy, either as an officer or as a director, assume the position immediately upon election to serve for the balance of the term (unless the person resigns, dies or is removed from office).

It would be possible to have an election for the person to take office at a later date.  With a vacancy in the office of president, normally a vice-president would be fulfilling those duties in the absence of the president, so it might make sense to have the new person take office at another time, perhaps at the next meeting of the members, for example.  But the purpose of a special election to fill a vacancy is to fill the vacancy, and usually to fill the vacancy as soon as possible after the election, i.e. immediately.