You can do it with Google's Geocoding API. What you're actually looking for is reverse-geocoding, but it supports that too. You can have it return XML and then parse it for the postal code (a.k.a zipcode). Should be quite simple actually.
To use the API, simply access it by HTTP with the correct latitude and longitude:
Then parse the XML using XPath, or your favorite XML parser (or just use JSON):
XPath xpath = XPathFactory.newInstance().newXPath();
String expression = "//GeocodeResponse/result/address_component[type=\"postal_code\"]/long_name/text()";
InputSource inputSource = new InputSource("https://maps.googleapis.com/maps/api/geocode/xml?latlng=37.775,-122.4183333&sensor=true");
String zipcode = (String) xpath.evaluate(expression, inputSource, XPathConstants.STRING);