Kyle here from Shopify,
The Checkout page is a secured page that needs to meet the requirement for processing credit cards, because of this it can't be edited.
For the Credit Card to be validated the Post Code, Country, Billing Address all have to be filled in. I do realize you may have an issue, but perhaps you can put a note on the Cart Page for your users to use a generic Postal Code if they do not know theirs. You can do this by going to your Online Store > Theme > click ... > edit html > cart.liquid from there you could add some text for your customers.