Thanks, fisicx. The radius script wouldn't be as useful to me as it would include locations within 5 miles as the crow flies but well over that to drive to (out in the countryside). I had already found http://www.postcode.org.uk/, which looks very promising to me, but my web designer still thought it couldn't deliver my requirements.
I've now sent him an example page from Deliveroo that first asks for the delivery postcode then allows or prevents (with a suitable message) completion of the order form, which is basically what I want, with the additional option of proceeding with an order on a booked collection basis where the postcode doesn't qualify for delivery.