No, they do not! If a store doesn’t support Apple Pay or Google Pay, you can still use the Zip app to shop there - the merchant will just need to enter the card information manually.
After selecting the In-Store tab in your Zip app, tapping Pay in-store, and entering how much you plan on spending, you will need to select Pay with Apple/Google wallet to create your in-store Zip virtual card.
Then, go back to the In-Store tab in your Zip app and you will see the card number, expiration date, and security code of your Zip virtual card. Show this to the merchant cashier and they can then manually enter this information in their system for your in-store order.