...

$searchString = isset($_GET['q']) ? $_GET['q'] : null;

$users = User::getByLimitedDomains();

if (!empty($searchString)) {
    $users->search(
        function($user) use ($searchString) {
            /** @var User $user */
            return preg_match('/(.*)'.$searchString.'(.*)/i', $user->getEmail()) === 1; // case insensitive search
        }
    );
}

...