New vanity light won’t work?

If you are showing power at the light and it will not work check the neutral wire in the switch box. It may have come loose. All white neutral wires should be connected together under a wire nut in the back of the switch box. 

If you just bought it, then you can return it. Also, you can return it with the customer support, saying it is broken. If you are really tech savvy, you can try to fix it yourself, don't if you don't know how to do it. The place where you bought it will surely let you return it, if it doesn't work.

check every thing is working with a multi meter

is there a hidden switch somewhere? multimeter test in the sockets will demonstrate that there is power. and did you test bulbs in an ordinary lamp, first?

Am I right in thinking these are Edison Screws? Sometimes the shape of the lamp can prevent it screwing in sufficiently to make contact on the end.

Are there any shade rings that you can remove?  they often conflict with the shape of the lamps especially if the orig design was for golf balls or some such.