
However - what if person A orders a delivery in the name of person B (who knows nothing of the transaction) and Domino's person arrives to find no-one waiting? Not that that's likely to happen, of course...

Why not designate one or two people (even members of staff who are on duty) to phone the orders and ensure that the delivery is collected (from one delivery point) and paid for?
That way, nothing can go wrong, there would be no security issues and no-one would waste away through malnourishment.
By the way, did you know that even the kebab shops in Horsham now deliver?