I think it somewhat depends on where you are staying, but I don't think a concierge upgrade at no charge is very likely to happen. We checked into the WL last May and were asked if we wanted to upgrade to concierge at an extra cost of $35 a night. That is a LOT cheaper than if you had booked your trip in advance through CRO and requested concierge. Well, in the end we didn't opt for it. I have seen similar instances on the boards here, either people were approached by a CM or sometimes as a guest they asked at check-in. I have heard varying rates between $30-$40 a night.
That being said, I think that is when they are pretty slow, and I'm not sure if all the hotels do this. I wouldn't count on it being available. If you were thinking of it, and didn't want to commit up front for extra $$$, you could try asking at check-in to see what becomes of it, just know that it may not be available when you get there.
Good luck!