This is actually a tricky question. Using a geocoding solution like GeoNames is likely to lead to major errors for a lot of queries. The reason for this is that GeoNames by looking up the record in their database that is closest to your query point and then returning the ZIP code they have on record for that point. This works great when your query point is right on top of a record in their database, but can lead to errors otherwise. For example, if their nearest record is a few blocks away in a different ZIP code, you'll get the wrong answer.
The US Census Bureau has created maps of the ZIP codes:
https://www.census.gov/geo/reference/zctas.html
Please see their notes on that page.
I have also worked on a project that uses the Census maps to provide an API that gives back the ZIP code for a given latitude and longitude. It is at:
http://askgeo.com
We offer both a web API and a Java Library that you can run on your own server. The library has excellent performance. Since our site offers additional information than just the ZIP code, you can read about our ZIP code service here:
http://askgeo.com/database/UsZcta2010
And you read about the documentation for the Web API here:
http://askgeo.com/#web-api
The GeoNames methodology is fundamentally flawed for this type of query. If you are looking for the polygon that contains a given query point, you need a map with the polygons, and you need a spatial index to provide fast look-ups. GeoNames has neither. AskGeo has both.