Add a [PROMO_CODES_USED] shortcode to the EE messages, this shortcode outputs any promotion codes used within a transaction. If the Promo code is a coded promotion the format will be "Promo Name[{Promo_code}]" otherwise its use "Promo Name".

tw_add_promo_shortcode.php

<?php
/*
Plugin Name: Event Espresso site specific functions
Description: Add custom functions for Event Espresso to this plugin.
/* Begin Adding Functions Below This Line; Do not include an opening PHP tag as this sample code already includes one! */


function tw_ee_register_new_promo_shortcode( $shortcodes, EE_Shortcodes $lib ) {
  
   if ( $lib instanceof EE_Transaction_Shortcodes ) {
      //Add your shortcode to the add as the key, the value should be a description of the shortcode.
      $shortcodes['[PROMO_CODES_USED]'] = _('This shortcode outputs any Promotion codes used during registration.');
   }
   //Return the shortcodes.
   return $shortcodes;
}
add_filter( 'FHEE__EE_Shortcodes__shortcodes', 'tw_ee_register_new_promo_shortcode', 10, 2 );
function tw_ee_register_new_promo_shortcode_parser( $parsed, $shortcode, $data, $extra_data, EE_Shortcodes $lib ) {

   //Check we are within the EE_Transaction_Shortcodes
   if ( $lib instanceof EE_Transaction_Shortcodes && $shortcode == '[PROMO_CODES_USED]' ) {

      //Pull the transaction from the EE_Messages_Addressee object passed to parser.
      $transaction = $data->txn;

      //Check we have an EE_Transaction object
      if ( $transaction instanceof EE_Transaction ) {

         //Pull in the promotion line items for this transaction
         $promo_rows = EEM_Price::instance()->get_all_wpdb_results(
               array(
                  array(
                     'Promotion.Line_Item.TXN_ID' => $transaction->ID()
                  )
               ));

         //Setup an arrey to store all promo codes used on the transaction
         $promo_codes = array();

         //Loop through promo line items and build the promo_codes array using the Promocode name and code (if available)
         foreach( $promo_rows as $promo_row ) {
            if( $promo_row[ 'Promotion.PRO_code' ] ) {
               $promo_codes[] = sprintf( '%1$s [%2$s]', $promo_row[ 'Price.PRC_name' ], $promo_row[ 'Promotion.PRO_code'] );
            }else{
               $promo_codes[] = $promo_row[ 'Price.PRC_name' ];
            }

         }

         //Implode the promo_codes array into a comma-delimited string.
         $promo_codes = implode(',', $promo_codes );  

         //Return a single string or promo codes used.
         return $promo_codes;

      }
   }
   //If not within the correct section, or parsing the correct shortcode,
   //Return the currently parsed content.
   return $parsed;
}
add_filter( 'FHEE__EE_Shortcodes__parser_after', 'tw_ee_register_new_promo_shortcode_parser', 10, 5 );