Our association is facing the same problem. Our president resigned at the beginning of this month.
Our bylaws do not cover resignations, so I have to go to the
Florida Statutes, which only covers the situation where a replacement is needed to constitute a quorum. In such a case, any homeowner can force the issue by filing an Intent To Apply For Receivership. This gives the association a 30-day time limit to fill the vacancy to avoid the large costs incurred by a court-appointed receiver.
Since ours was a 3-member board, the 2 remaining do constitute a quorum so I'm not sure there is anything forcing them to fill the vacancy, except that common sense dictates that a tie-breaker should be added.