Dropbox isn't meant to be used silently as a datastore behind an application, as made clear by requirements such as:
Step 2 of authentication. Applications should direct the user to /oauth/authorize. This isn't an API call per se, but rather a web endpoint that lets the user sign in to Dropbox and choose whether to grant the application the ability to access files on their behalf. The page served by /oauth/authorize should be presented to the user through their web browser. Without the user's authorization in this step, it isn't possible for your application to obtain an access token from /oauth/access_token.