I would think you would need to sand it down to a very nice furniture finish and then re-urathane the surface. I would contact a trusted carpenter or wood shop for the best products to use.

How does the water get on the wood? Was there a leaky faucet? Or other users that didn't wipe up splashes?

Can you tell if the damage is simply a discoloration to the finish itself, or has water indeed gotten through the finish and spotted/stained/damaged wood?

Do you know what type of finish was used on the vanity when it was built?

Some finishes can be renewed. A cleaning, perhaps a deglossing to take the sheen down, then a recoat.

Other finishes need to be removed to bare wood and then refinished with something similar or something new, along the lines of what enduring is referring to.