This turned out to be a permissions issue due to the way PowerChurch checks for folder access rights.
When My Documents is redirected to the server, by default, users are allowed to access their own My Documents folder, but not to see the list of all users' folders.
When PowerChurch checks for the My Documents folder, it does so by first going to its parent folder, then checking to see if My Documents exists before changing to the My Documents folder. With folder redirection, since normal users are not allowed to list the folders in the parent folder, an access violation occurs.
I was able to work around the problem by going to the parent folder on the server and granting the List Folder permission to authenticated users.
It would still be nice that if opening My Documents fails for some reason, PowerChurch would fall back to another folder, perhaps the PowerChurch program directory, rather than aborting the program. It would also help if the dialog would remember the folder from one time to the next. (We'll always be attaching newsletter PDF files out of the same folder.)