Login to Dropbox from browser extension on Firefox

I can't offer help on the Firefox side of things, but in case you can't use a redirect URI, it is worth noting that a redirect URI is actually optional on the Dropbox OAuth app authorization flow. That is, the use of a redirect_uri on /oauth2/authorize is not actually required; it's optional when using the code flow. If it is not supplied, the user is presented with the authorization code directly on the Dropbox web site after authorizing the app, which they would need to manually copy/paste into the third party app to complete the flow. While the user experience is certainly not as good, that may be an alternative if needed.