ShoppingList is a small, independently made app. If something isn't working, or you'd like to try it, the fastest way to reach the developer is directly by email.
ShoppingList is currently distributed through TestFlight while it's in testing. Email the address above and you'll be added as a tester.
Location is used to detect when you've left a store you've set a location for, so ShoppingList can prompt you to confirm what you picked up. That detection needs to work even when the app isn't open, which is why iOS asks for "Always" rather than "while using the app". See the privacy policy for the full detail — location never leaves your device.
Sync happens through your iCloud account. Check that you're signed in to the same Apple ID on both devices, and that iCloud Drive is turned on in Settings. If it's still not syncing after a few minutes, email support and mention both device models and iOS versions.
The order isn't set manually — it's rebuilt from the order you check items off, every time you check or uncheck something. If you backtrack for something you forgot, that gets learned too, which can look odd for a trip or two until it settles. If an order still looks stuck or wrong after several trips, email support with the store name and what you'd expect instead.
Email a short description of what you did and what happened instead of what you expected. Screenshots help. A general description of your device and iOS version is useful too.