How can i add organization Legal name to search?
How are you letting the users lookup their employer?  Is that an autocomplete existing user on the webform?

If so, you can customise what information is displayed in the autocomplete, but that does not currently include legal name.  I'd guess that is a trivial enhancement.

If you want the option of the autocomplete searching on legal name, that might be a bit more effort.  For both, you could raise an issue at https://www.drupal.org/project/issues/webform_civicrm

To add an option for legal name search in the normal search would need a core change following the same pattern as nickname.

A different approach would be to provide a separate lookup facility on legal name using a view but that is not as integrated.