Ohio Rev. Code § 3501.07
At a meeting held not more than sixty nor less than fifteen days before the expiration date of the term of office of a member of the board of elections, or within fifteen days after a vacancy occurs in the board, the county executive committee of the major political party entitled to the appointment may make and file a recommendation with the secretary of state for the appointment of a qualified elector. The secretary of state shall appoint such elector, unless the secretary of state has reason to believe that the elector would not be a competent member of such board. In such cases the secretary of state shall so state in writing to the chairperson of such county executive committee, with the reasons therefor, and such committee may either recommend another elector or may apply for a writ of mandamus to the supreme court to compel the secretary of state to appoint the elector so recommended. In such action the burden of proof to show the qualifications of the person so recommended shall be on the committee making the recommendation. If no such recommendation is made, the secretary of state shall make the appointment.
If a vacancy on the board of elections is to be filled by a minor political party, authorized officials of that party may within fifteen days after the vacancy occurs recommend a qualified person to the secretary of state for appointment to such vacancy.
R.C. § 3501.07