Run feeds importer from outside the site
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.