No councillor, chairman or otherwise, has authority to spend money or indeed to do anything unless specifically authorised to do so by resolution of the council. However, some authorities allocate a "chairman's allowance" to their chairs which is designed to cover any additional expenses associated with the role. HMRC take the view that this is a "benefit" and should be paid via the payroll system and taxable unless it is only used for expenses such as travel (to outside meetings, where applicable), stationery, etc.
I've noticed that one local council nearby has a "chairman's petty cash". I've no idea where the authority is for that!