I have found a niche in the delivery business, would to exploit, looking for advise on the most economical technology to accomplish this.
An automated system could easily accomplish this. Human call centers would be slow and costly for something like this, if I understand correctly.
I can think of a way of testing this idea without building any technology. You can setup a toll free number with extensions for each driver's cell phone. After placing an order the customer will receive the extension for their driver should they need to be in touch. Actually, in Europe drivers delivering packages are frequently in touch with customers to determine their current location and hand over the package. There they simply use cell phones without any real backend infrastructure.
That said, what is the pain point here that you would be solving? If the customer feels they need to contact the driver, that would likely mean that the driver is either late and/or potentially lost. Both of those things would be better fixed at the root of the problem by getting a better driver/GPS.
Happy to discuss further if you'd like.
It depends on what sort of communication you'd like to have between the driver and customer. SMS is a passive route that requires the least effort from both parties and can have automatic event triggers (e.g. you're pizza is on its way)or direct communication in short bursts. I've had good experience with Twilio as a provider for SMS messages.
Sounds similar to how the Uber app works. When you order an Uber cab, you can follow the driver on the map, and communicate with him. I would start your search by figuring out how Uber does it :)