I realise that the Dropbox app has dropped support for iOS 9. But it also seems that you can no longer login via the Dropbox website (I tried several browsers with no luck). The 'signin' button is always greyed out.
This means my app users can no longer use Dropbox at all on iOS 9 since they cannot link the app to their Dropbox account.
Can you please confirm whether there is a way round this on iOS 9?
Yes, that's correct, given that both the official Dropbox iOS app and the official Dropbox web site do not support iOS 9, the Dropbox OAuth app authorization flow is also not supported on iOS 9. (Technically, you might be able to implement a system where the user processes the authorization flow on another computer/device and somehow delivers the access token to the iOS 9 device, but that would need to be custom-built.)