What can I use to remove coffee cup stains from my marble vanity top?

Try a paste of baking soda and water, in a circular motion.

https://www.youtube.com/watch?v=7vv7S9Yzm-0

Mr Clean Magical eraser will remove coffee stains. It is supposed to be safe for marble, but if you have any doubts, you may want to call a home improvement store or even contact Mr. Clean Magical Eraser hotline personally to ask for a list of safe materials on which to use it. Use gloves.

You need something that will gently pull out the stain. Whiting powder (Calcium Carbonate) may work. dry powder is applied and let sit overnight. May require a lot of applications. Marble is very porous, you need to be very careful with it, even after applying a sealer.

I've never tried it, but I heard that toothpaste and baking soda will remove the stain