Since version 7.x-2.0-beta3, Feeds has Drush support. You could use the following command to trigger an import (where 'my_importer' is the machine name of your importer):
drush feeds-import my_importer
If the importer is attached to a content type, specify also the feed
node with the option '--nid':
drush feeds-import my_importer --nid=2
Note that a source for the importer in question must exist*. So you need to have gone through the import form at least once.
You could write a shell script that executes the drush command and that is triggered when the url is requested.
*The drush command 'feeds-import' has more options, allowing you to perform an import even when no source was created yet. But I don't think that applies in your situation. See https://drushcommands.com/drush-7x/feeds/feeds-import
for more information.