When you click to sign in, Delivery Status attempts to open Safari so it can show you a sign in page. Unfortunately this doesn’t always work correctly. You should be able to sign in manually like this:
There’s also a new “Amazon - by tracking link” option that lets you track deliveries without signing in. If you go to Amazon’s “Your Orders” page you can copy the link of any “Track package” button there and paste it in Deliveries (or use the Add to Deliveries extension on that page). This will track a specific shipment, rather than the whole order. In most cases these links are available as soon as the order is placed—you don’t have to wait until it ships.