Not being aware of any requirement to do so as a council is only beholding to the auditors for annual governance of finances and compliance procedures.
You are a democratically elected body with powers. One is election of a chair and second is employment of a clerk.
Your chair can be elected at your next PC meeting ( or EO meeting if you feel it urgent). As the chair's only function is to chair the meetings of the PC ( and represent the council in public) or if specified in standing orders deputise for the clerk in their absence I can see your only stumbling block. Perhaps a locum clerk to hold the fort at least until you elect a chair may be a solution but with cost implications( check with your county association)
So organise an EO meeting and elect a chair and get the employment of a new clerk moving.