wp_editor embed shortcode not working within my plugin

The WordPress OEmbed should show in the editor by default.

If it doesn’t, I suspect that you need to enqueue some scripts and styles on your page where you are using wp_editor. Eg:

wp_enqueue_script('editor', false, false, false, true);
wp_enqueue_script('quicktags', false, false, false, true);
wp_enqueue_script('wplink', false, false, false, true);
wp_enqueue_script('wpdialogs-popup', false, false, false, true);
wp_enqueue_style('wp-jquery-ui-dialog', false, false, false, true);
wp_enqueue_script('word-count', false, false, false, true);
wp_enqueue_script('media-upload', false, false, false, true);
wp_admin_css();
wp_enqueue_script('utils', false, false, false, true);