Looks like a good mail form processor. I would modify the error message to a variable format that way your flash form will recieve that message and you can act accourdingly.

Rework this part
Code:
function print_error($reason,$type = 0) {
   build_body($title, $bgcolor, $text_color, $link_color, $vlink_color, $alink_color, $style_sheet);
   // for missing required data
   if ($type == "missing") {
      if ($missing_field_redirect) {
         header("Location: $missing_field_redirect?error=$reason");
         exit;
      } else {
      ?>
      The form was not submitted for the following reasons:<p>
      <ul><?
      echo $reason."\n";
      ?></ul>
      Please use your browser's back button to return to the form and try again.<?
      }
   } else { // every other error
      ?>
      The form was not submitted because of the following reasons:<p>
      <?
   }
   echo "<br><br>\n";
   echo "<small>This form is powered by <a href=\"http://www.dtheatre.com/scripts/\">Jack's Formmail.php ".VERSION."</a></small>\n\n";
   exit;
}