They sell wood filler that you can use to fill in the dings, sand it smooth, and then stain over it and apply a coat of polyurethane in a finish that matches the rest of the vanity (matte vs shiny, etc). The trick will be finding a stain that matches the color of the rest of the vanity so that the patches won't be noticeable.
It depends on the wood. If it was real hardwood, perhaps a dampening then using a warm iron might bring the surface dent back up. If it is a vanier over particleboard - good luck - filling as already suggented and refinishing might be te best plan.