The App Store and the Wallet App are 2 different things. That the App Store is asking for a valid payment method is not directly tied to the Wallet App.
You can if you have a credit card added to the Wallet App use it for the App Store, which is why it suggests you add one to the Wallet app, but you can totally use the Wallet App without a credit or debit card added and still add a credit or debit card to your Apple ID for App Store and iTunes Store purchases.
If your Apple ID is requiring a payment method, then you must provide one, there is no way around this, but again this has no bearing on how you use the Wallet App. You can add the payment method for the stores without affecting the Wallet App at all by following the steps here: link->Change, add, or remove Apple ID payment methods - Apple Support
Once its been added and verified it can be removed and should not ask for it again.