Somehow I don't think so. I believe the game has to send data to the chair regarding which direction it should move and when to execute the movement. As far as I'm aware, it doesn't do that.
hrmm. I've got a bit of electronics hobbyist experience. If I were going to build something like this (and maybe this is how it works) I'd probably create a "pass-thru" in which you plug your stick into the chair and then plug the chair into your computer. This would allow the chair to eavesdrop on the stick data (which is a communication standard) and then move the chair in response to your stick input. Software on the PC could adjust the stick "sensitivity" and it would then work any time the stick is moved... Additionally, you could also have the software monitor the running games (effectively disabling the movement if the stick is accidentally bumped when there's no game in progress...)
I may build something like this myself when I move back to the states and have an actual garage. My small Japanese style apartment won't cut it!
[EDIT]
I may actually build a miniature mockup of something like this using some servos... that is if I can tear myself away from the game long enough to do anything "productive."
[/EDIT]
Tim