Hmm, you can extract objects which have a zipcode but I think in the USA the zip codes are not mapped as such. In Germany you have polygons which have postal_code, so all objects inside that polygon would have the same postal_code unless they have there own tag.
Such a polygon is a list of nodes and each node has a latitude and longitude.
I think in the USA there are no such polygons, so you might want to compute them somehow. As a starter try this overpass query
http://overpass-turbo.eu/s/Fvl
Note that some areas are poorly mapped while others look quite complete.