Yes, ShipStation can send tracking information to your customers if you enable the shipment notification emails in the store's Notification settings and your customer's email address is present on the order.
ShipStation's email notification option is off by default since most selling channels also send shipment notifications to your customers. ShipStation automatically notifies your selling channel when an order ships. This notification includes the tracking information as well, which your selling channel can pass on to your customer. However, you may want to use ShipStation's notifications instead so you can maintain control of all of your shipment notifications in one place and use ShipStation's notifications templates and customization features.
For manual orders, rather than orders imported from a store, you must add your customer's email address to the order. As long as the email address is present when you create the label, ShipStation can also successfully send the shipment notification to your customer.
Amazon and certain other selling channels do not allow third-party apps like ShipStation to send email notifications directly to customers. For these selling channels, the email notification options will not be present in the store's Notifications settings.