What sort of storage boxes are you looking for? Is it just a one time order?
If you’re only looking for a couple hundred cardboard boxes, you could talk to warehouses around the city (Tesco’s, Sainsbury’s, etc) and ask if they have any that they’re just going to throw out. I forget what company it was, but we got a great response from them and they practically threw them at us.
I remember hold boxes for customers back in the day, most boxes are crusted as soon as they are put out, so best ask in the morning and collect them in the afternoon. FYI, fyfe banana boxs are the best (strong).