This is not an uncommon model. And while it doesn't dovetail into the vanilla PO/INV models, it can be accomodated.
The key thing to keep in mind is that your salespeople are writing sales orders not purchase orders. Do they receive a PO# from their client when they write the order (Helpful, but not crucial for reconciling invoices)? You could implement a system for your scenario easily - and even integrate it with a platform like Ariba (Or not, and simply do point to point).
If you have reps get a signature for a delivery rather than just dropping an invoice you will actually have a much more accurate system that your clients will appreciate. Sounds like you already provide a huge level of customer service - couple that with very fast (More accurate) invoicing, POR's and you've got a great system.
Hope that helps, happy to expand further.
Director, Channel Programs Hubspan Inc.