Will the Nikkor AF-S 55-200mm VR lens auto focus on Nikon D5100?

I know that the Nikon D5100 doesn't have a built-in focus motor so I was wondering if I use a lens that has an auto focus option will that work?

AF-S means that the lens has a built-in motor, so, yes.

Any AF-S lens provides full functionality on your camera.

Yes, it will.

Any lens marked with "AF-S" will focus on a D5100 or any other Nikon camera that does not have a built in focus motor.

Any lens that has AF-S written on it will, also if you ever buy something from a third party lens brand make sure they have a focusing motor like the hyper sonic focusing from sigma or equivalent from tamron

Yes it will auto focus. Any AF-S will auto focus with your camera.