The live address is missing from the Customer Account API settings of your Headless storefront.
The symptom. Customers can log in on your .webflow.io address, but on the live domain the login fails or comes back to an error.
Why. Shopify’s customer accounts send the customer back only to the addresses you list.
The fix. In Shopify, open the Headless storefront’s Customer Account API settings. Add your live address, such as https://www.example.com, to the JavaScript origins and to the Callback URIs, next to your .webflow.io address, each without a slash at the end.