Your query is in single quotes so it won't pass the variables.
Change
$sql = 'UPDATE `legacy` SET `date` = \'$date\', `agent` = \'$agent\', `aemail` = \'$aemail\', `prod` = \'$prod\', `acctemail` = \'$acctemail\', `custname` = \'$custname\', `custemail` = \'$custemail\', `custphone` = \'$custphone\', `custreq` = \'$custreq\', `resexp` = \'$resexp\', `authcc` = \'$authcc\', `mgr` = \'$mgr\', `completed` = \'$completed\', `comdate` = \'$comdate\', `resolution` = \'$resolution\' WHERE `legacy`.`id` = \'$display_id\' LIMIT 1;';
to
$sql = "UPDATE `legacy` SET `date` = '$date', `agent` = '$agent', `aemail` = '$aemail', `prod` = '$prod', `acctemail` = '$acctemail', `custname` = '$custname', `custemail` = '$custemail', `custphone` = '$custphone', `custreq` = '$custreq', `resexp` = '$resexp', `authcc` = '$authcc', `mgr` = '$mgr', `completed` = '$completed', `comdate` = '$comdate', `resolution` = '$resolution' WHERE `legacy`.`id` = '$display_id' LIMIT 1";
It get's rid of all the nasty looking backslashes too.
You should also pass all the $_REQUEST variables through mysql_real_escape_string() to prevent SQL injection
e.g.
$display_id = mysql_real_escape_string($_REQUEST['display_id']);