I'll second Serendipity. If there is someone willing to take on the duties and the board approves, then let them do it. If you can't find someone willing to fill the spot, maybe you could find someone willing to do it until the new school year starts and then you could recruit/elect a new secretary.
Since you have no bylaws there is nothing to dictate how you resolve this issue. If you have someone that is willing to take the job and the remaining board members agree on this person then it is okay to put this person in as the secretary.
Since it has happened to you already, make sure you cover this matter when you create your bylaws over the summer.
Please help, Our newly elected secretary is leaving the school this summer. How do we go about putting a new secretary in place before the new school years starts in August? Can the remaining board members appoint someone or do we have to wait until school starts for an election?
We have never had by-laws in this school, that was our summer project so I have nothing to go by.