The fuel pump is mechanical, not electric, and runs off a camshaft in the engine block. The cam spins and moves a lever on the pump. Fuel comes in, is pumped through check valves, and up to the carb. A glass cover lets you see the condition and operation of the pump.
We did not suspect a bad fuel pump when working the kinks out of the carb. Fuel would spray out of the supply pipe on the carb. But there were air bubbles in the fuel when we cranked the engine, indicating air was leaking in and lowering our fuel pressure. It was also dirty.
I had read horror stories about removing the pump from the block. Access on the front bolt is tight. The right tools make a huge difference: small 1/2" socket on a 1/4" drive and the narrowest extensions you can find. With that setup everything lined up and it was a piece of cake removing it. One thing that annoyed me was that there is no drain on the pump for the glass cover so be prepared to spill an ounce of fuel when removing the cover.
Reinstalling is tougher as you're fighting the tension of the lever on the cam while trying to line up the bolts and not let the gasket drop.
I removed and reinstalled the pump about 4 times and can now do it in under ten minutes.
Winter 2019 Update
In November-December 2019 I started having start-up issues after the Jeep sat for a few weeks. I think the system doesn't seal out moisture very well and water was able to mix with the fuel. I added new fuel and Heet dry gas and was able to cycle through fresh gas by running the engine on a can of starting fluid. I think the lesson is not to let the Jeep sit for too long without a warm-up. For more detail, see my post here: