I assume your bylaws don't have anything in them relating to this type of situation? I know ours don't.
I guess I wouldn't see that the board would have to be dissolved. Couldn't they just operate as a group? The board really should be working together as a team anyway, the President is just there to lead them along. It might be difficult without a designated leader, but I have always found that in any group there is someone who takes on that role even without the title. This would allow your group to continue with planning for next year.

No! I approached them both individually for the Pres position and then tried for Co-Pres. One agreed to be VP and then the night of elections they decided to be Co-VP but almost pulled out since there was no Pres and they do not want that position. My real question still is, Is there a board if no one is going to step up for the title?

Would the V.P.s be willing to serve as co-Presidents? I did this with a friend a couple of years ago. We divided the duties so we didn't have us both working the same issues. It would depend on whether your bylaws allow for that too. It can be a good situation with the right personalities and clear division of labor, but can be tough if there are control issues. Even if they would be willing to handle the post over the summer months and the nominating committee could continue to seek a President to take the reins by the beginning of the school year it would be helpful.

We just held our elections last night and we still have no president. We have 2 new co-vice pres that have made it clear that they will not be pres. Is there a board without a president?
The nominating committee really droped the ball, only 1 of the 3 even bothered to contact anyone about running. I'm the one that did most of the calling & talking to try to find a new Pres. I think they tought one of the new VPs would feel guity and step up. I'm the current VP and if I wasn't moving away this summer I would be the Pres next year.