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
For example, to upload a file, you can use:
https://www.dropbox.com/developers/documentation/http/documentation#files-upload
By the way, that's a link to the documentation for the HTTPS endpoints themselves, but we recommend using one of the official SDKs if possible:
https://www.dropbox.com/developers/documentation
Those have corresponding native methods for the HTTPS endpoints.
