Shopify has now resolved the incident with zipcodes being passed to the checkout which in turn has allowed Delivery Rates by Zipcode to correctly return rates at the checkout. Thanks for your patience.
The issue is being caused by a global issue from Shopify not any Zapiet servers, you can read more about it here
https://status.shopify.com/incidents/t8d3s0bfk91t
We are monitoring this closely and will post updates as we receive them.
We are currently investigating an issue where Delivery Rates by Zipcode may not return shipping rates in the checkout. We are actively working to analyze the cause of the issue and get a fix in place. More updates to follow.