So, there is a way to generate the URL required to load the branded tracking page using the data from the API...
You can get your branding ID/page ID from any of the links Shipment and Delivery notifications shipstation sends to your clients when the Branded Tracking Page is enabled under Store Setup.
I've asked support about this as well and was told that no, there currently is no way to get a link to the branded tracking page. I don't remember the exact wording of why they told me it wasn't a thing, but it was along the lines of the branded link that is sent to the customers in the emails is tracking number specific. Idk how difficult it would be to set up a tracking page that pulls the information from ups (or whatever carrier you're using) when the tracking number is entered into the tracking field, and display it the same way it does when you click the link in the email notification.