To accomplish this you have two things to deal with.
First, I assume those accessing the site from outside have a different URL than the ones accessing the site from inside. You can deal with this by adding two lines to wp-config.php. This tells WP, and this plugin, to use the URL accessing the site as the site and wordpress URLs.
define( 'WP_SITEURL', $_SERVER['REQUEST_SCHEME'] . '://'. $_SERVER['HTTP_HOST'] );
define( 'WP_HOME', $_SERVER['REQUEST_SCHEME'] . '://'. $_SERVER['HTTP_HOST'] );
Second, you want existing posts to have relative URLs so they display properly both inside and outside the organization. There are a couple of ways of handling this. If there are only a few posts at the moment, just enable this plugin, edit and re-save each post. This will convert the URLs to relative. The second option is to use the related_sites filter in functions.php so that the plugin treats specific URLs as though they are relative.
add_filter( 'of_absolute_relative_urls_related_sites',
function( $related_sites ) {
$related_sites[] = "https://originalsite.org";
return $related_sites;
}
);
Replace originalsite.org with your own domain.
Hope this helps,
Andrew P.