I'm not sure if this is still valid for you, but we recently released vimeo-networking-java which should help with OAuth authentication. The specific instructions on how to get it working can be found here in the README.
But the basic idea is that we provide you with a method to take care of the code grant redirect and it can be implemented with the following:
private void onLoginClick() {
// Go To Web For Code Grant Auth
String uri = VimeoClient.getInstance().getCodeGrantAuthorizationURI();
Intent browserIntent = new Intent(Intent.ACTION_VIEW, Uri.parse(uri));
startActivity(browserIntent);
}
If you follow the rest of the readme, it will cover how to get the redirect uri set up.