ยง 2113. Filling  vacancy  in  office  of  trustee. 1. A vacancy in the
  office of trustee in any school  district  may  be  filled  by  election
  within  ninety  days  after  it  occurs.  If not so filled, the district
  superintendent of the supervisory district, within which the schoolhouse
  or principal schoolhouse of the district  is  situated,  may  appoint  a
  competent person to fill it.
2.  If  a  vacancy  in  the  office  of trustee in a union free school
  district exists, the commissioner  of  education  may  order  a  special
  election for filling such vacancy. When such special election is ordered
  the vacancy shall not be filled otherwise.
3.  If  such  vacancy is filled by a district meeting, it shall be for
  the balance of the unexpired term; but when such vacancy  is  filled  by
  appointment  by  a  district  superintendent, it shall be only until the
  next annual meeting of the district.
4. A vacancy in the office of trustee in union  free  school  district
  number  one  of  the  town of Tonawanda, Erie county, shall be filled by
  appointment by the remaining members of the board  to  serve  until  the
  next annual election at which the vacancy may be filled.
Structure New York Laws
Title 2 - School District Organization
Article 43 - School District Officers--Town and County Officials
2102 - Qualifications of Officers.
2102-A - Required Training for School District Officers.
2103 - Ineligibility to Office.
2103-A - Police Officers and Firefighters on Boards of Education.
2106 - Terms of Officers of Newly Created District.
2108 - Notice and Acceptance of Election.
2109 - Refusal of Trustee to Serve.
2110 - Penalty for Refusal to Serve or Perform Duty.
2111 - Resignation of District Officers.
2113 - Filling Vacancy in Office of Trustee.
2114 - Filling Vacancy in Office of Clerk, Collector or Treasurer.
2115 - Notice of Appointment to Fill Vacancy and Filing Thereof.
2116 - District Records, Books, etc., Are District Property.
2116-A - Books and Records to Be Kept by School Districts.
2116-B - Internal Audit Function.
2118 - Expenses of School District Officers.
2119 - School District Trustee Not to Draw Draft on County Treasurer in Certain Cases.