The governing body must elect a chair and a vice-chair from among their number. There are no specific regulations about the actual conduct of the elections but they should comply with the principals of fair and democratic elections.
Please ask your clerk to the governors if you need more advice on this. Other important points that governing bodies (GB) need to adhere to are:
Where a vacancy arises in the office of the chair or vice chair, the governing body must elect one of their number to fill that vacancy at their next meeting.
Where the chair is absent from any meeting or there is at the time a vacancy in the office of the chair, the vice chair is to act as chair for all purposes. If the vice chair is not in attendance the governors in attendance at the meeting should vote in a chair for the meeting (governors employed by the school cannot take the chair).
More help like this can be found in the DFE Governors Handbook or The School Governance (Roles, Procedures and Allowances)(England) Regulations 2013.