Fire webhook after static export is done.

functions.php

<?php

add_action('ss_after_cleanup', function(){
  // Get options from Simply Static.
	$options       = get_option( 'simply-static' );
	$basic_auth    = $options['http_basic_auth_digest'];
  
  // Prepare URL and data.
  $url  = 'https://test.com';
  $data = [];

  $response = wp_remote_post( $url, array(
      'body'    => $data,
      'headers' => array(
          'Authorization' => 'Basic ' . $basic_auth ),
      ),
  ) );
});