When an officer resigns, dies, becomes incapacitated, or removes residency from the District, such office shall be considered vacant. When a vacancy occurs in any office, other than the office of Moderator, Clerk-Treasurer, and Collector of Taxes, the Prudential Committee shall forthwith, by appointment, in writing, fill such a vacancy until the election of a successor at the annual meeting next thereafter. At such annual meeting, the voters of the School District shall fill the vacancy in any elected office, other than the office of Moderator, for the remainder of the term. When the office of the Moderator, Clerk-Treasurer, or Collector of Taxes becomes vacant, the Prudential Committee shall forthwith by appointment, in writing, appoint a Moderator, Clerk-Treasurer, or Collector of Taxes to fill such vacancy until midnight of June 30 following election of a successor, in the case of the Moderator's office, or appointment of a successor, in the case of the Clerk-Treasurer's office or the Collector of Taxes' office. All appointments under this section shall be filed and recorded in the office of the Clerk-Treasurer. No person shall be appointed to fill a vacancy in office who would not be eligible for election to such office.