The advanced search form provides a dropdown for the periodical title and the value it passes into the _GET array is the index selected, which corresponds to the id for that periodical in the database. I needed to modify the $_fieldMap variable in vpn-search/classes/VPNFormBuilder.php so that the publisher field in the form mapped to the organs.org_id field in the DB.
The other possible approach would have been to change the optionsFromSql('organs', 'or_id', 'or_name') call in the form_advanced.php file to optionsFromSql('organs', 'or_name', 'or_name'), but as some of the organ names contain spaces, that might cause problems. I have not tested that.
The other odd thing is that I'd like to use the "organ-name" controls in the form on the page, but I can't seem to get the correct behaviour when I use controls with those names. I must be missing something obvious, as identical code (except for the name of the input element on the page) produced different results.