Step 1
You may have a bar-light type fixture for your vanity light. Inspect to see if any bulbs are burned out or loose. Even if a bulb appears out, it may just need to be screwed in to make it work. You can shake bulbs that are not working to determine whether or not you can see or hear a loose filiment. If so, just replace the bulb. Make sure not to exceed the manufacturer’s maximum bulb wattage or you may cause a fire. This bulb rating is usually stated on a small foil sticker located inside each bulb socket. If this doesn't solve your problem, move on to the next step.
Step 2
Use your electrical tester for light sockets and check each socket to see if they are getting power. Usually, you attach one end to the center of the socket node and the other end can touch the metal side. Read the tester instructions for exact directions. If the light sockets are not getting power, move on to the next step. If the light sockets are getting power, move on to the next step or you might need to replace the light fixture.
Step 3
Use your electrical tester with audio beep and check power at the switch. Remove the switch cover with the screwdriver and remove the switch itself as well. Hold your tester to each of the two terminals and make sure you are getting power. If the wires are hot, but the switch is not working, then replace it. Just color match the wires to the new switch and hook it up. If this doesn't solve your problem, go to the next step.
Step 4
Remove the light cover by using your screwdriver. Carefully pull the electrical wires out of the back of the base. Use your audio tester to see if it beeps, meaning power is present. If so, check for loose connections. Replace the wire nuts if needed. If this doesn't get your light going, you may need a professional electrician.