What kind of bulbs does this lamp need?
I do not think this is a light bulb connection as is and shown in the picture. Without the cap removed and having the ability to see what's behind it, I do not think it may be answered. Look at the photo below for the different light bulb connections.

 

NOTE: Whoever gave the thumbs down did me wrong. When I posted my answer the first 2 picks were all that was posted and and the text was rewritten after showing those are screws in the fixture. I simply did not remove my answer because I was not sure if we are supposed to.

This type of light bulb (connectors) is referred to as Double Contact Bayonet light bulbs.

My bedroom ceiling lights all use them and they can be found at most hardware stores like Home Depot, Lowes, and even Walmart.

Removing the light bulb: Push in and turn counterclockwise. When it stops turning it can be pulled out. About a 1/16 of a turn. (opposite for replacing the bulb.)