Dropbox offers a Java API for core features, you can access Dropbox file content with it: https://www.dropbox.com/developers/core/docs
If the user does not need to retrieve the files directly you can also use the datastore API which allows you to store your application Data: https://www.dropbox.com/developers/datastore/tutorial/http
For both APIs you need to create an application key. The user has to visit the dropbox site once, afterwards you can work in the background. A simpler alternative would be a file export, the user could choose the sync directory of any cloud service he desires.