How to Give Author/Users the right to embed in WordPress

The capability you are after is called unfiltered_html. Some options:

1. Modify author capabilities in your theme functions.php. This is saved in the DB, so you can access a page, make sure it works then remove it from your functions.phpfile. A better option would be to run it on theme activation. See this page on WP Codexfor options:
   

2. function add_theme_caps() {
       // gets the author role
       $role = get_role( 'author' );
   
       // This only works, because it accesses the class instance.
       // would allow the author to edit others' posts for current theme only
       $role->add_cap( 'unfiltered_html' ); 
   }
   add_action( 'admin_init', 'add_theme_caps');