Upload & Sell: Off
Accounting is simple; paper trail. There is NOTHING wrong with paying or being paid in cash as long as there is something on both parties side saying there was a payment and there was a receipt. If they want an invoice it's cool to have too.
It aint rocket science.
OP, this should have been made clear prior to you agreeing to shoot. You need to know how and when you will be paid. A request for an invoice is fine but it should be noted prior to for your own sake. A lot of businesses want an invoice....even accountants. Maybe even have an invoice ready just in case. I'm sure the hired photographer would be rather shocked and come up with another reason to not pay right then and there.