Not as part of the import, but after the import, a script can loop through the found set of records produced by the import and use Insert From URL to insert the file into a container field. Import your URL into a text field, BTW, not a container field.
Thanks PhilModJunk! That was easy to set up. My script ended up being:
If [IsEmpty (my_container_field)]
Insert from URL [Select; my_container_field; my_container_field_url]