(a) When any local or general law authorizes or requires the remaining members of the governing body of a political subdivision of this state to appoint a person to fill a vacancy occurring in that body, the absence of a quorum caused by any such vacancy or vacancies shall not impair the authority of a majority of the remaining members of such body to make that appointment.

(b) When any local or general law provides for the appointment, other than appointment by the Governor, of a person to fill a vacancy in an elective public office and such vacancy is not filled within 45 days after the vacancy occurs, the Governor shall appoint a person to fill such vacancy. A person appointed by the Governor to fill a vacancy pursuant to this subsection shall serve until such time as that person would serve had the person been appointed as provided in that local or general law.