Sidebar endpoint using WordPress API

Is there a default endpoint for sidebars?

No, I don’t think there is. So (for now), using a custom REST API endpoint does sound like a good option to me.

gives me the sidebar not in json format and then a json response with an empty body_response

That’s because get_sidebar() echo the output, hence the $response_body is empty.

So you’d want to use output buffering like so:

ob_start();
get_sidebar( 'one' );
$response_body = ob_get_clean();