Use Registered Username for Author

if (is_user_logged_in()) {
    // the user is logged in, so display the submission form
    if (function_exists('user_submitted_posts')) user_submitted_posts();
} else {
    // the user is not logged in, so redirect to any URL and exit
    header('Location: https://example.com/');
    exit;
}

As the FAQ says: https://www.ads-software.com/plugins/user-submitted-posts/faq/

Yes this is an excellent example of why it’s important to always read the plugin documentation (e.g., readme.txt) before using any WP plugin.