That depends on the requirements of the state in which the corporation is incorporated.

First, the concept of an "S" corporation is a tax concept only. I do not know of any state that imposes special board requirements based on whether the corporation has elected to be taxed as an "S" corporation.

Second, exactly what board positions must be filled (chairman, vice chairman, secretary, etc.) can vary state by state. The states with which I am familiar generally impose requirements for a minimum number of board members. However, those states have a special rule under which the board need never have more members than the corporation has shareholders.