Only one of the bulbs will light. I got a continuity checker out and found one part of the ballast which does not have continuity. I don't know where I would get a replacement part and don't even know what kind of part it is.

I'm guessing since there are two of the same items and one does not have continuity that is the problem. I've included a photo. The part with no continuity is located directly above a metal housing with a copper winding in it and it has a red wire going over it, In front of it there is a resistor, on each end of the small metal tube is a solder connection with one lead to the winding in the housing the other leading to where the bulb mounts the resistor is connected between it.

Can someone identify the part and tell if a replacement would solve the problem and where I would get one from?

Honestly, these fixtures cost $11.00 at the big box hardware stores, and cheaper and easier to buy a new fixture, rather that locate and buy a new ballast kit, which is what you took a photo of, and what is defective.

Honestly, these fixtures cost $11.00 at the big box hardware stores, and cheaper and easier to buy a new fixture, rather that locate and buy a new ballast kit, which is what you took a photo of, and what is defective.

Thanks! Going to recycle what I can and move on to next project.