Yes you can. As long as you have enough clearance inside your vanity and the vanity is constructed out of material that would allow for a hole to be cut without compromising it's integrity, you can install BleuBox the same way you would inside a wall. We highly recommend that you consult a professional to make sure the material your vanity is made of is able to support the weight of BleuBox.
BleuBox has a lacquered finish that will not tolerate harsh chemicals. We recommend cleaning with soap on a soft, damp cloth, then dry completely.