Would having two co-presidents sharing the position of president be legally construed as being in violation of our bylaws and endangering our 501(c)(3) status?

First, the IRS doesn’t care what you call your officers or how many you have, and the presence or absence of a president will, by itself, have no effect on your 501(c)(3) charitable exempt status.

Second, it is not unusual to have vacancies among officer positions and I wouldn’t consider it illegal to have a vacancy.  Most state nonprofit corporation laws have a requirement for a person to function as president, but your bylaws probably allow the board to appoint “other” officers so the board could appoint two people to serve as president -- co-presidents -- and spell out their rights and responsibilities.  That is a little messy, but I think not illegal.

Whether it is a good idea is another question.  I was involved with a nonprofit 25 years ago that tried to have co-presidents, until a major funder came along and said that an organization with co-presidents didn’t have a strong enough board to be worth funding.  If you don’t have somebody willing to take leadership responsibility, it does say a lot about the strength of your board.  While you temporize with co-presidents, you ought to do some serious board development to find people willing to commit to the organization.