§ 22-18-101 — (b) The governing body, by ordinance, shall specify the procedure for determining whether a vacancy exists

(c) If a vacancy is determined to exist, the governing
body shall appoint an eligible person to the office who shall
serve until his successor is elected at the next general
municipal election and qualified. If the entire council is
vacant, the district judge for the district in which the city or
town is located shall appoint a person to fill each vacancy and
serve until the next general municipal election at which time a
successor shall be elected to fill the unexpired portion of each
term.
(d) A vacancy in the office of mayor shall be filled only
from the governing body.
(e) Vacancies in appointive offices shall be filled in the
manner provided for initial appointments.