The chairman can step down at any point.
The first agenda items on the next meeting of the parish council will be to elect a new chair (this is the law). If the (former) Chair remains as a councillor, he will chair the election and have a casting vote.
There is no requirement / obligation for it to the vice-chair.
However, in the absence of there being a Chair, and until a new Chair is elected at the next meeting, the vice-chair has the same rights and responsibilities as the chair (N.B. legally the vice chair has their rights even when there was a chair in place). - Anything that can be done by the Chair can be done by the Vice Chair - except where standing orders prohibit such action.