If the electors of any incorporated town shall fail to choose a mayor, councilman or auditor, provided that such office exists, or if any person elected to such office shall neglect or refuse to serve therein, or if a vacancy shall occur in the office by death, resignation, removal from the incorporated town, or otherwise, a majority of the remaining councilmen may appoint a successor who is a registered voter and upon their failure to make such appointment within thirty days after the vacancy occurs, the vacancy shall be filled within fifteen additional days by the vacancy board. Such board shall consist of the town council and one registered elector of the town, who shall be appointed by town council at the council's first meeting each calendar year or as soon thereafter as practical and who shall act as chairman of the vacancy board. If the vacancy board fails to fill the position within the time prescribed, the chairman shall, or in the case of a vacancy in the chairmanship, the remaining members of the vacancy board shall, petition the court of common pleas to fill the vacancy. In the case where there are vacancies in a majority of town council, the court of common pleas shall fill such vacancies upon presentation of petition signed by not less than fifteen registered electors of the town. In all cases, the successors so appointed shall hold the office until the first Monday in January after the first municipal election occurring more than sixty days after the vacancy occurs, at which election an eligible person shall be elected for the unexpired term.