Order number + email required

For example, German legislation — specifically §356a BGB, which implements the corresponding EU directive — explicitly requires that the consumer must be able to provide information identifying the contract they wish to withdraw from. In practice, this identification can only be ensured reliably by using the order number or the invoice number.
By the way, the consumer’s name is also mandatory.

(2) Die Widerrufsfunktion muss dem Verbraucher ermöglichen, eine Widerrufserklärung an den Unternehmer zu übermitteln und dem Unternehmer in oder mit der Widerrufserklärung ohne Weiteres folgende Informationen bereitzustellen oder zu bestätigen:

1. den Namen des Verbrauchers,
2. Angaben zur Identifizierung des Vertrags oder des Teils des Vertrags, den der Verbraucher widerrufen möchte,
3. Angaben zum elektronischen Kommunikationsmittel, mit welchem dem Verbraucher eine Eingangsbestätigung für den Widerruf zu übermitteln ist.

More complete example, because setting the form field to “required” does not trigger any validation by the Ajax callback. In this example, all fields first name, last name, order number in addition to e-mail address are mandatory to be filled. Action eu_owb_woocommerce_process_order_withdrawal_request will be called by order_withdrawal_request and provides access to already processed form data stored in $meta . Bear in mind provided data is not semantically checked at all; here you could build on existing shop checks to validate provided content.

// make order_number mandatory by adding required flag to args
function aaa_eu_owb_make_input_field_required($args) {
        if (is_array($args)) {
                $args['required'] = true;
        }
        return $args;
}
add_filter('eu_owb_woocommerce_form_field_order_number_args', 'aaa_eu_owb_make_input_field_required', 10, 1);
add_filter('eu_owb_woocommerce_form_field_first_name_args', 'aaa_eu_owb_make_input_field_required', 10, 1);
add_filter('eu_owb_woocommerce_form_field_last_name_args', 'aaa_eu_owb_make_input_field_required', 10, 1);

// check if provided data is non-empty; we expect order_number, first_name, and last_name to be mandatory.
function aaa_eu_owb_check_mandatory_fields($order, $error, $items, $meta, $was_guest) {

        $err_msg='';

        if ( empty( $meta['order_number'] ) ) {
                $err_msg .= ', '._x('Order number', 'owb', 'woocommerce-germanized' );
        }
        if ( empty( $meta['first_name'] ) ) {
                $err_msg .= ', '._x('First name', 'owb', 'woocommerce-germanized' );
        }
        if ( empty( $meta['last_name'] ) ) {
                $err_msg .= ', '._x('Last name', 'owb', 'woocommerce-germanized' );
        }

        if (!empty($err_msg)) {
                $error->add( 'missing_fields', _x( 'Bitte prüfen Sie Ihre Angaben: ', 'owb', 'woocommerce-germanized' ).substr($err_msg, 2));
        }
}
add_action('eu_owb_woocommerce_process_order_withdrawal_request', 'aaa_eu_owb_check_mandatory_fields', 10, 5);

Thanks @vdwoocommercesupport – great idea to make fields configurable in the backend. I can confirm that it works like expected.