Should any person appointed to membership of any board of equalization fail or refuse to take and subscribe to the oaths required by Section 40-3-3 within five days after being notified of his appointment to membership on said board, or shall for any other reason herein enumerated become disqualified to serve as a member on said board, or should a vacancy on any board occur from death, resignation, or any other cause, then the Commissioner of Revenue shall select from the remaining nominees on the list from which such original nominee was taken a person to fill the position to which such disqualified person was first appointed or to fill the vacancy on said board. The proper nominating body may, however, make additional nominations so that the total of its nominees from which appointments may be made to fill vacancies shall be brought up to a total of not more than three such nominees. Should the remaining nominees be, for any reason, disqualified to serve on said board, then the Commissioner of Revenue shall request the proper nominating body to name three additional persons eligible to serve on said board, and from such additional nominees a person shall be selected to fill the vacancy occasioned by the disqualification of the original nominee or nominees or other vacancy. Any appointment to fill a vacancy on any board of equalization shall be for the unexpired term only.