I can find a bug report that says exporting container field contents from Web Direct doesn't work, but the results described differ from yours.
For More Information see: WebDirect Export Field Contents Inconsistent with FileMaker Pro
This is one of many acknowledged bugs that can be found in the Known Bug List thread here in the Report an Issue section of the forum.
It can also be downloaded as a database file from: https://www.dropbox.com/s/jt09b82i0xijbu3/FMP%20Bugs.zip
That doesn't sound like the same issue. The fact that I'm getting a "file not found" error makes me suspect that the web direct module does is not able to access the directory where the container contents are stored.
Yes, but even after you resolve that issue, this may not work until they release a bug fix.
Can the web user see anything in the container field?
For use with Web Direct, the container field should have external storage specified and the fields should not be inserted with the "store a reference" option specified.
Yes, from a web browser I can see the files that are in containers and I can even upload files to containers using drag and drop or by right clicking and inserting. The containers are are not set to store as references. I've tried both the external data storage option (both secure and open) and the default option where it stores it as part of the database. All get the same result.