Here's what happened after following Coke's advice:
My wife called her family in Sarajevo, who went to the airport with the ID card. In the meanwhile, we checked in at Lisbon. The ground staff would not check her through to Sarajevo, but offered to check us to Vienna.
After we cleared security, we received an e-mail message from the manager of the Austrian office in Sarajevo, saying that she had the ID card, that she had spoken with the Granična Policija to confirm that my wife would be allowed to enter with it, that she would meet my wife at the plane to give her the ID card, and that she had noted this in our reservation record.
We were unable to get boarding passes for the last leg despite this. We are awaiting takeoff to Munich and will try again there. Our layover in Vienna is only 40 minutes, so the best case scenario seems to be that we will get on the plane and our luggage will not. Fingers crossed....
... And, on arrival in Munich, where we had just over an hour, we went to the gate, where the gate agent issued boarding passes for the last leg. She assured us that our bags, originally checked to Vienna, would be sent to Sarajevo.
We made it, though our luggage did not.
The flight to Vienna was about ten minutes late. The flight parked at one of the bus spots rather than going to a jet bridge, and the airline sent a very competent fellow with a small van to take us two, along with one other person with a tight connection, to immigration and then directly to our next flight. We never even found out its departure gate. We certainly didn't have time to inquire about our bags.
The flight left about a half hour late, and I was hopeful about our luggage when the pilot announced that part of the reason for the delay was the delayed loading of some luggage.
As we walked from the plane to the arrivals hall (as one does in Sarajevo; it always makes me think of the 1950s or '60s), we found the staff member with the ID, whereafter we went inside and passed through immigration quite normally.
The luggage, as mentioned, did not arrive, so we filed a lost luggage report. It occurred to me that we didn't discuss with the agent in Munich that the bags were originally checked only to Vienna, and that she might have said they'd go to Sarajevo on the basis of an incorrect assumption. Or maybe the bags really didn't have time to make the transfer. I suppose we may never know.