The office of a mayor or councilmember shall become vacant if the person who is elected or appointed to that position fails to qualify as provided by law, fails to enter upon the duties of that office at the time fixed by law without a justifiable reason, or as provided in RCW 35A.12.060 or 42.12.010. A vacancy in the office of mayor or in the council shall be filled as provided in chapter 42.12 RCW. An incumbent councilmember is eligible to be appointed to fill a vacancy in the office of mayor.