????? Not clear what the wires in the first photo are connected to. Not clear what the cable in the second photo is connected to or its location/height on the wall. Are the wires in the first photo connected to the lights in the third photo? You called the cabinet a "vanity". A vanity would be a cabinet that would holds a sink and countertop. Is this some kind of medicine cabinet? Confused.

Sorry, good to clarify. It is a medicine cabinet- my bad. First photo is back of medicine cabinet. Second photo is wire coming out of wall where it wired.

How do the wires in your first photo connect to those in the second? The proper way would be with a junction box.

They just connected and pulled through the wall. If we added an old work box in the wall, would that be enough even though there isn’t a ground wire on the medicine cabinet.

No, leaving the ground wire unconnected in an old work box does nothing for safety. Those metal trim rings around each light socket need to be grounded, that's your safety issue. On the back side of those sockets, behind the wood board, someone needs to connect a grounding wire to the metal frame of each socket holder and connect them back to the ground wire from the power supply. Instead of an old work box mounted in the wall, I would mount a box in the backside of the cabinet to contain all the electrical connections.