Hi Eben,

Dropbox exposes an API which you can call via standard http POST calls https://www.dropbox.com/developers/documentation/http/overview. You can use the REST web service call functionality from Mendix

Another approach would be to add the dropbox Java SDK https://github.com/dropbox/dropbox-sdk-java to you rMendix project and expose the functionality you need as java actions.

I believe the first approach is fast and dirty and you should only use it if you only have a few API calls. If you need a more extensive integration then I would recommend (and so does Dropbox) to use the java library, because it will save you a lot of trouble down the road. 

Dropbox quote: We recommend that you use one of our officially supported SDKs e.g. Java instead of raw HTTP calls.

Hope this helps,

-Andrej