Question

Can a Parish council have a vice chairman if there is not an elected chairman

Answer 1

It's hard to see how this could arise.  At the annual meeting each May, the first item of business is the election of a chairman.  The councillor elected as chairman remains in that position until either they resign or the next annual meeting comes round. If they resign, then the next council meeting would normally elect a new chairman as the first item of business.

Further research since the above was written indicates that a council is not properly constituted without a chairman, and the first item at a meeting of a council that has no chairman must be the election of the chairman.  The council cannot function without a chairman.