Hello Stephane, and thank you for posting to our community.
ShippingEasy adds a note to the order in WooCommerce with the fulfillment information (carrier, service, tracking number, etc) but no update is made to the order data itself because WooCommerce does not provide any place within their standard order data for storing shipment tracking data.
The fulfillment information should, however, be available to the WooCommerce Shipment Tracking plugin, which is used by WooCommerce in the notification information (order summary page, emails, etc) that are sent/displayed to your buyers.
If you are not seeing that information being made available to your buyers, can you please contact our support directly using the support links in the top left of your account? It would be helpful if you could include screenshots for any affected WooCommerce orders, displaying the notes section on the right side as well.
Thank you and happy shipping!