Is there a way to see CiviMail logs through the interface (v. 4.6.36.)
If you look under Administer - System Settings - Extensions and click on the Add New tab you'll see one called CiviCRM Log Viewer. Install that and then admins will be able to view the logs using the user interface.
Since you're on an old version of Civi you may need to download an older version of the extension from https://civicrm.org/extensions/civicrm-log-viewer, but if you don't have access to the file system then I'm not sure how you'd install it. Tricky.