(a) Except for the office of township or multi-township assessor, if a township fails to elect the number of township officers that the township is entitled to by law, or a person elected to any township office fails to qualify, or a vacancy in any township office occurs for any other reason including without limitation the resignation of an officer or the conviction in any court of the State of Illinois or of the United States of an officer for an infamous crime, then the township board shall fill the vacancy by appointment, by warrant under their signatures and seals, and the persons so appointed shall hold their respective offices for the remainder of the unexpired terms. All persons so appointed shall have the same powers and duties and are subject to the same penalties as if they had been elected or appointed for a full term of office. A vacancy in the office of township or multi-township assessor shall be filled only as provided in the Property Tax Code. For purposes of this subsection (a), a conviction for an offense that disqualifies an officer from holding that office occurs on the date of (i) the entry of a plea of guilty in court, (ii) the return of a guilty verdict, or (iii) in the case of a trial by the court, the entry of a finding of guilt.