Well, I have narrowed the problem down to an actual bug in the JVM. Here is a link to it:
Bug Database According to them, this issue has been fixed. But it hasn't. Also, on a related bug, they say it works fine in j2sdk1.4.2. Must be nice to be able to install a JDK release that isn't available yet and say everything is fixed. That doesn't help people that need it to work now.
The solution is you have to setTraversableCycleRoot(true) on the JApplet root container. Then you have to setFocusable(true) on every component you need to tab to. This is a workaround and does work.
However, there is still a problem. When creating my own KeyListener I was trying to capture the Tab KeyEvent.VK_TAB. This still doesn't work. In doing this, I noticed that hitting ENTER, SPACE, and TAB all give the same KeyEvent Integer of 0 for some reason. So they are not registering themselves correctly or something.
Anyway, just wanted to give an update as to the information I have found our regarding this problem.