Storynory and AudibleKids are two sites that offer audio stories for download. They are built around a common premise but do have some notable differences.
Storynory features only free downloads so their collection is focused on classic stories, fairytales and even some original works that haven't been published elsewhere. You can subscribe to the site through iTunes in order to download their stories more expeditiously or just pick and choose the files you want. They have organized their stories into different categories and you can subscribe to the feed from individual categories as well.
AudibleKids has a few free titles but most of their content is for purchase only. They have a much broader selection of titles ranging from well-known picture books for younger kids on up to chapter books for middle schoolers. You are more likely to find up-to-date titles on this site and can expect to spend anywhere from a few dollars up to, and over, twenty dollars per audiobook.
Both of these sites could be used to bolster your school or classroom's listening library. Of course, another option would be to have your students record their own audio books by using something like GarageBand or Audacity and build your listening library organically from within your own school.