There is no statutory requirement to have a Vice Chair, but it could be argued that if your Council has created the position of VC and your VC has resigned, they should be replaced. If you expect your Chair to be unavailable for the forthcoming meetings, I would elect a new VC, hopefully someone with sufficient support to continue in the role after the Annual Meeting.
As an aside, has your Chair given apologies for each meeting and have those apologies been accepted and minuted by the Council? If not, the six-month rule under Section 85 may come into play.