How can I use two bathroom vanity lights without backs?

It looks like there are screw holes in along the top of the light. I would tack a small piece of wood to the wall that would go inside the light and screw down through those holes into the piece of wood. That should hold it just fine.

Yes, that is a simple fix, all you need is

A. 2 round steel boxes (for wiring to a Junction), 1 steel Junction box (to wird the two lamps to a switch) a Switch of your liking and 2 Peices of Wood or Metal or Anything really that strike your fancy to moung those lamps on.

Or

B. If you want to however plug those in independently of a wall switch you will need a Lamp Kit and a Bag of Wire Nuts

Disassembly is pretty much straightforward.

Been doing this since age 8. Trust me, I fixed more lamps than God wiped his a**. Easily 8000.

You basically unscrew each Lugnut that is holding the lamp to the mirror and then you can mount the same lamp through anything with a hole.

Attaching to 1 or 2 Lamp kits is probably your safest bet.

That is straightforward too. White is the Ground. Black is the Live or Hot Wire.

Exactly what GandmasHouseDIY suggests. Screw a board to the wall and screw the fixture to the board through the top holes.

Nice fixture. You didn't say if you had an existing light where you want to mount this.

You guys are awesome! I love the simplicity of mounting a piece of wood and attaching the light to the wood. What a great idea. I had an electrician come out to the house to do it and he refused saying it wouldn’t meet code charge me $50 for a trip charge and left. I have been flabbergasted ever cents LOL thanks again

Allison,

It does not meet code because you have loose wires that could come in contact with a combustible surface (your wall). While the risk is extremely low, there is a small risk of fire should the fixture become overheated or short out. If you had the back on, it would contain the wires and wire nuts. There might have been a small opening in the back with a plastic or rubber bushing where the wires would come out of the back and be wired to the junction box in your wall. I would suggest using LED bulbs. They generate much less heat. Also, buy a small piece of foil backed duct insulation and cut it to fit in the back of the fixture. It's very similar to the type of insulation used in light fixtures. A couple of strips of electrical tape will keep it in place. Keep the wires between the insulation and the front of the fixture except where they have to come out to be wired to the junction box. Where the wires come out, wrap them in electrical tape. That should be plenty of protection.