That’s an interesting question! Unfortunately the API is just for accessing local notes.
You can get around it by using the CLI, this will allow you to initiate a sync from the CLI and the access all the up-to-date notes using the API.
If you’re familiar with JavaScript it might be possible to extract out the Dropbox access stuff and make your own API, or it could be possible to write your own in another language (but there is no spec so maintanaince would be a burden). If you go that route please let me know, I’d like to try it out!