If it’s not built into the core REST API, then likely not. I’d post a bug or feature enhancement in Trac and see if you can get some feedback there.
They allowed me to do it with this code:
//allow orderby=menu_order
add_filter( 'rest_post_collection_params', 'my_prefix_add_rest_orderby_params', 10, 1 );
function my_prefix_add_rest_orderby_params( $params ) {
$params['orderby']['enum'][] = 'menu_order';
return $params;
}