You haven't explained the specifics (of the rate and so on).

I booked a reservation at Avis a few weeks ago, and then two days before pickup I cancelled it. Nothing has been charged. I use an airline AWD and "pay later".

While I have a credit card on file with Avis (as I'm Avis Preferred [Plus]), there was no credit card specifically with the reservation.

So was yours a "pay now" reservation? Most Avis reservations (that aren't "pay now") don't ask for a credit card, so how can they bill you?