Dropbox does offer an API you can use for listing, uploading, and downloading files, among other operations. You can find everything you need to get started with the Dropbox API, including documentation, tutorials, and SDKs here:
https://www.dropbox.com/developers
Note that this is intended for programmers who are looking to write their own code for their integration. If you're not a programmer, you'll want to instead look for a pre-built solution from a third party.
