Woocommerce function to display this custom field value on the order edit page

I am using your code in addition I used the woocommerce_before_order_notes hook to add files and working fine for me.

/**
 * 
* VAT Number in WooCommerce Checkout
*/
add_action( 'woocommerce_before_order_notes', 'wpdesk_vat_field' );

function wpdesk_vat_field( $checkout ) {
    echo '<div id="wpdesk_vat_field"><h2>' . __('Nakup na podjetje') . '</h2>';
    
    woocommerce_form_field( 'vat_number', array(
        'type'          => 'text',
        'class'         => array( 'vat-number-field form-row-wide') ,
        'label'         => __( 'V kolikor nakupujete kot podjetje, vnesite ID za DDV' ),
        'placeholder'   => __( 'Vnesite ID za DDV' ),
    ), $checkout->get_value( 'vat_number' ));
    
    echo '</div>';
}
add_action( 'woocommerce_checkout_update_order_meta', 'wpdesk_checkout_vat_number_update_order_meta' );
/**
* Save VAT Number in the order meta
*/
function wpdesk_checkout_vat_number_update_order_meta( $order_id ) {
    if ( ! empty( $_POST['vat_number'] ) ) {
        update_post_meta( $order_id, '_vat_number', sanitize_text_field( $_POST['vat_number'] ) );
    }
}
add_action( 'woocommerce_admin_order_data_after_billing_address', 'wpdesk_vat_number_display_admin_order_meta', 10, 1 );
/**
 * Display VAT Number in order edit screen
 */
function wpdesk_vat_number_display_admin_order_meta( $order ) {
    echo '<p><strong>' . __( 'ID za DDV', 'woocommerce' ) . ':</strong> ' . get_post_meta( $order->id, '_vat_number', true ) . '</p>';
}
add_filter( 'woocommerce_email_order_meta_keys', 'wpdesk_vat_number_display_email' );
/**
* VAT Number in emails
*/
function wpdesk_vat_number_display_email( $keys ) {
     $keys['ID za DDV'] = '_vat_number';
     return $keys;
}

Tested and works.

Checkout

Edit Order

Mail