First, I would not say that this is in violation of 501(c)(3) regulations because the Internal Revenue Service does not enforce the provisions of state law which control the governance of the association.  The IRS cares about what the association does, and not so much about how it selects its officers or directors.

It is normal, and probably specified in your bylaws, that the vice president will assume the role of the president if the president is not available to act.  The bylaws and the state nonprofit corporation law also probably state that officers and directors serve for their specific term and until their successors are selected.  Therefore, the vice president, regardless of what he or she may be called, has authority to perform the duties of the president.

State nonprofit corporation laws also recognize that nonprofits occasionally fail to hold an election at the proper time.  That’s one of the reasons they state that officer holders shall hold their offices until their successors are selected.  The organization does not stop functioning because it doesn’t follow its internal rules.

But the nonprofit corporation laws normally also say that if the required elections are not held within a specific period after they were supposed to be held, the members who have the right to vote in the election have the right to call a special meeting at which to hold the election. Check your state law to see if it has such a provision, and if so, do the members care enough to force a new election?  You may find that they are more than happy to have someone willing to serve several more years.