How do I use OAuth within my Java GWT application? In particular, I want to get a list of users in my Google Aps domain, using this API:
http://code.google.com/googl
For integrating OAuth and GWT, you should start with Scribe which handles the implementation of the OAuth:
https://github.com/fernandezpablo85/scribe-java
Next, you need to create a GWT widget that can handle the user's interactions to acquire permission to access their account. Then grab the response token, and make the API requests to the external site.
No point re-implementing OAuth when scribe already does it for you - you just need to. I'd probably aim to use a GWT Popup for doing the authentication:
http://gwt.google.com/samples/Showcase/Showcase.html#!CwBasicPopup