Whenever a customer places an order an email will be sent to the customer about the order confirmation. Thereafter if an order is fulfilled with tracking number, an email will be triggered to the customer.
Just wanted to explain about an Auto Fulfill app, if it can meet your needs.
Auto Fulfill app does order management and Auto Fulfillment of orders. When you receive an order, it will be notified to your vendor by email and your vendor will ship the products and enters the tracking number in the vendor link.
The order gets fulfilled automatically in Shopify and your customer gets notified by email with a tracking number.
You also get notified every time an order is fulfilled.
Let me know if you have any questions.
Here is the link for the app: https://apps.shopify.com/auto-fulfill
Auto Fulfill-Order Management & Auto Fulfillment App
To send the "Order Shipped" status along with the tracking number, you have to fulfill the orders along with the tracking number and by selecting a carrier (see the image below). Once you fulfill the order, your customers will receive an email stating that "Order is on the way" along with the tracking number and a link to the order status page.
Once your customer clicks on the "View your order", it will redirect to the Order status page where they can see the tracking number.
Now, if you want to display the Status of you order on the "Order Status" page, you will have to fulfill your orders with any of these four carriers: FedEx, USPS, UPS or Canada Post. Also, Shopify will send the "Order out for delivery" and "Shipment delivered" notifications when you fulfill orders that have tracking information from these four carriers.
For other carriers, you will have to use an app like Shipment Tracking & Notify app, which will help you send tracking notifications to your customers. The app also allows you to have an integrated tracking lookup page, from where your customers can track their orders.