Would like to store files in Dropbox and automatically sync them to be displayed (as a file list) in my Drupal 7 website. There is a Drupal module that is supposed to do this (https://www.drupal.org/project/dropbox_client) but it does not seem to work.
That module is made by a third party, so we can't offer any support for it.
Dropbox does offer an API that enables programmatic listing, uploading, and downloading of files though, so this is possible. You can find everything you need to get started with the Dropbox API, including documentation, tutorials, and SDKs here: