Well, I thought I'd follow up to let you all know what happened. We had to go back and run the cable anyway because they also needed their POTS lines up there that they had forgotten about. If running one cable, why not run two, right?
The husband was at work (the one who insisted upon no holes) today, but the wife called and begged us to come out and do the work anyway. I knew that we could fish everything in with zero exposed wire, but this guy wasn't having it. Anyway, we got it done, no exposed wire, no holes, no blank plates, nothing. Thank goodness for vinyl siding...
She was elated. A triplex face plate with her two POTS lines and a network jack connected to the Verizon router. Exactly what we should have been able to do yesterday.
As we were going to leave, I noticed that the IP phone was working. Apparently, he went out to Best Buy last night and purchased another Linksys router (not an access point), got it programmed and had Internet working on the desktop PC and the IP phone. The existing Verizon router was still in place in the basement, so I'm sure he didn't use two APs as outlined in the link that Trace provided. I don't know what he did, but I verified that it worked by making a phone call.
Although we ran a second cable for network use, we didn't end up using it for now. Two POTS lines and we were out of there. Thanks for the information in case I need it next time.