Is there any way to view the order itself from the original website (specifically beyond the immediate order completed landing page)? Might be difficult if you did guest checkout.
Did you get an order confirmation from both the online seller and from shop?
Have you tried using the shop.app website itself, from a desktop computer? I have no experience with it, and there’s a possibility it will try to still make you download the app after you sign in with the email that the order was sent to.
Try contacting the support email of the original seller for the tracking number?