dashboard widget form not submit mails

There are three issues I see in your code:

1. Your form doesn’t have an action set. You should add it and set the value to admin-post.php.

   <form method="POST" action="<?php echo esc_url( admin_url( 'admin-post.php' ) ); ?>">
   

   Because the function which processes the form and then sends the email is hooked to admin_post_submit_support_ticket (i.e. admin_post_<action>) and that hook is fired only on wp-admin/admin-post.php.

2. In _submit_support_ticket(), the isset($_POST['support_ticket_hash']) should be ! isset($_POST['support_ticket_hash']) because if not, the page would exit even if there was actually a valid nonce.

   if( ! isset($_POST['support_ticket_hash']) || ! wp_verify_nonce( $_POST['support_ticket_hash'], 'submit_support_ticket' ) ){
     //echo '';
     exit;
   }
   

3. The email subject is actually empty and that will not send the email. So make sure that your “Request type” drop-down has valid options (currently, the option‘s value is empty).

   wp_mail( '[email protected]', '', 'Testing' );   // bad; subject is empty
   wp_mail( '[email protected]', 'Hi', 'Testing' ); // good; subject is good..
   

   In fact, if you look at wp_mail()‘s docs, the subject is a required parameter. So once again, make sure the <option> has a good subject, although it might be preferred to compose the subject in your PHP and not sent as-is as coming from the form.

   <p><?php _e('Request type'); ?></p>
   <select name="support_ticket_type">
     <option value="Support request"><?php _e('Support request'); ?></option>
     <option value="Modification request"><?php _e('Modification request'); ?></option>
   </select>
   

And after the email is sent, you should send the user back to the dashboard page to prevent them from seeing a blank page. So after the wp_mail() call, you could do:

wp_redirect( admin_url() );
exit;

Also, you should always sanitize user-supplied data. Never trust their input even if you actually trust the person…