I've been doing some light searching on this, and I don't have any real answers. I have to assume that since most flights are grounded, international cargo (and mail) isn't moving around much either. Of course I couldn't find anything about the U.S. Postal service (certainly not on its own website!) but I did find this link, which said, "Europe's largest mail and express delivery company Deutsche Post said it was switching to road transport where possible."
Of course road transport is not possible between here and Europe, so I'm settling in for an indefinite wait in delivery and arrival of a good deal of international mail. I've got quite a few correspondents in the UK, Finland and Germany in particular... and I guess those Postcrossing postcards I just sent to Germany and the Netherlands are going to hang up in limbo for a while, too.
What a fascinating and amazing turn of events, this volcano!
If anyone knows of any postal information for specific countries, do comment and fill us all in. (Thanks to a very helpful comment from Erasercarver, I can share this like to Royal Mail to learn what's going on in the UK.)