Agree with Delboy'swife. No one councillor or even the clerk can act on behalf of the council without authorisation from the council to do so. This is usually provided for in a scheme of delegation, which delegates the council's powers and duties to the clerk or other officers (eg RFO), and committees. Outside of that scheme of delegation, a resolution by council would be required to authorise the chair (or any other councillor or officer) to act. Typically, the power to sign documents to give effect to resolutions of the council is delegated to the clerk.
It should be remembered that the chair has no special powers beyond those of any other councillor, except the right and obligation to chair a meeting of the council when they are present, the right to call an extraordinary meeting of the council, and the right to call a parish meeting.
You should check your scheme of delegation - what does it say about any powers delegated to the chair?
Can you give more details about the documents you are being asked by the clerk to sign?