It's hard to describe Magento novice how to add new attribute to a store :( Although for an experienced developer it's not hard:
- show one more input field in templates
- create mysql upgrade to add one more field for a store
- overload standard store model and save this attribute along with store saving process
But there are good news! :) I think, that your problem can be solved much more easier - every store has Shipping origin address, that is set up in System -> Configuration -> Sales -> Shipping section section. So you can setup it for each of your stores and show zip code by that value.
The only case when this solution doesn't work - it's when shipping origin doesn't match your store's office location. In such a case you really need to add office address as store's attribute. If this is your case - I recommend you to hire some Magento developer at oDesk or somewhere else to describe and show you the process of adding, storing and retrieving these attributes. It really cannot be described in one text message at StackOverflow.