Whew. Just heard from the shop. The takeaway is that it can be fixed today for a not unreasonable sum.
If I'm understanding correctly, the housing that the oil filter screws into has a drain and the check valve there failed, letting it leak. The VW dealer in the town down the road has a replacement, so they're going to get their hands on it, replace it and fill it back up with oil (when I checked last night, the level was barely touching the bottom of the dipstick).
So glad that we're not going to have to shuffle our schedule/reservations for the trip home.