75 ILCS 5/4-4
Vacancies shall be declared in the office of trustee by the board when the elected or appointed trustee declines or is unable to serve, or is absent without cause from all regular board meetings for a period of one year, or is convicted of a misdemeanor for failing, neglecting, or refusing to discharge any duty imposed upon a trustee by this Act, or becomes a nonresident of the city, village, incorporated town, or township, or who fails to pay the library taxes levied by the corporate authorities. Vacancies shall also be declared in the office of trustee by the board when, at the election of the first board of library trustees or at any subsequent election, there are not sufficient trustees elected to fill an entire board of 7 trustees.
Vacancies in the board of trustees in a city or a village under the commission form of government shall be reported to the mayor or president and be filled in like manner as original appointments.
If a vacancy occurs in the board of trustees in any incorporated town or village (other than a village under the commission form of government) or in any township, the vacancy may be filled by the remaining trustees until the next regular library election at which library trustees are scheduled to be elected under the consolidated schedule of elections in the general election law, at which election a trustee shall be elected to fill the vacancy for the remainder of the unexpired term. If, however, the vacancy occurs with less than 28 months remaining in the term, and if the vacancy occurs less than 88 days before the next regular scheduled election for this office, then the person so appointed shall serve the remainder of the unexpired term, and no election to fill the vacancy shall be held. If there is a failure to appoint a library trustee or a failure to elect a library trustee, or if the person elected or appointed fails to qualify for office, the trustee may continue in office if available and qualified until his successor has been elected or appointed and qualified. Vacancies shall be filled within 90 days after a vacancy has been declared .
75 ILCS 5/4-4