Pizza places have maybe 20 ingredients in stock that all get used at a pretty predictable rate. Keeping something like a much larger selection of fruits available and fresh when you can't freeze them is a somewhat harder problem.
I don't believe inventory management is that crude that they can't do deliveries in short time. Given that supermarkets were doing it in an albeit slightly longer timeframe (weekly), then I don't see why they can't refine the process to be same day. If anything, it gets the stock out of the store more quickly and they make a quicker sale.