Gelato is a fully white-labeled solution. As such, we don't send any information directly to your end-customers.
When your orders are marked as shipped, we will forward shipping information along with the tracking number to Squarespace. Tracking codes and shipping confirmation emails can be sent to your end-customers automatically through Squarespace.
You can read more about tracking for your store in Squarespace's Help Center article Tracking a customer's order shipment.
You can also check the status of your orders as well as the tracking links in the Gelato Dashboard.