I just removed permanent marker from my kayak with AIO, so AIO is obviously capable of cleaning many types of surfaces. I also just AIO'd my wash bucket to remove some scuffs or grease stains of some sort.
So give AIO a try, and take before/after pics for us.
Try Poorboy's Pro Polish, AIO will work too, its not as good of a cleaner, but if you have it handy, why order something just for a chair. Unless, of course, you just wanted to try something new .