I can't be certain about this, but my impression is that a council credit card may fall outside a strict interpretation of the legislation.
However, a number of local councils are obtaining credit cards, and it is certainly possible - in the sense that banks will provide one.
The problem is that the main legislation covering local councils is the 1972 Local Government Act. Much in the act is still perfectly sensible, but some things have changed a good deal in the intervening forty years. Financial practices are one.
There are now quite a few things that are extremely difficult to buy except by use of a credit card, and it is unreasonable to expect the clerk to use their own financial resources for council matters.
So my view is that it is reasonable, as a matter of expedience, for a council to obtain a card, and for the clerk to use it, subject to strict controls (including use only for expenses that cannot be handled by cheque payments).
Much the same applies to variable direct debits.