Travis - maybe I can help you understand the accelerator pump.
This is a VACUUM accelerator pump. There is no direct linkage from the throttle to the pump. The pump operates as follows:
(A) when the engine is running, engine vacuum is applied to the vacuum cylinder in the top section of the carburetor.
(B) the vacuum pulls the top vacuum piston up into the cylinder. Some models have an adjustment screw that will allow adjustment of the depth the upper piston is pulled into the cylinder; some do not.
(C) as the upper piston is pulled into the upper cylinder, the operating spring is compressed.
(D) when the operator increases the RPM of the engine, vacuum decreases (normally goes to zero under wide-open throttle conditions).
(E) as the vacuum decreases, the stored energy of the compressed spring overcomes the force of the decreased vacuum, allowing the lower piston (the pump) to be forced down into the well, and pumps fuel into the pump jet to be sprayed into the throttle area.
Note that since there is no direct connection from the throttle to the pump, pumping the footfeed prior to attempting to start the engine results ONLY in the exercise of the operator's ankle. A vehicle with an updraft carburetor should ALWAYS be cold-started using the choke. With a warm engine, no choke should be necessary.
Rusty is incorrect about buying another carb to obtain a pump. Pumps are available, but expensive. You do NOT wish to purchase one except as a last resort.
Rusty mentioned that this is a Zenith 63AW series. Assuming he is correct, the body of the carburetor will be cast iron. After removal of the float (soldered) and the venturi (diecast); heating the pump well with an acetylene torch will help to free a stuck piston. The principle is quite simple. The piston is stuck because of corrosion (oxide). Heating the outside of the pump well will heat the corrosion and burn the oxygen from the molecules of corrosion; thereby reducing the space occupied by the corrosion molecules. This will help the penetrating oil to penetrate.
And the Zenith 63AW series is one of the finest updraft carburetors ever made!
Oh, and one caveat! Try a magnet on the carburetor to make sure it is cast iron BEFORE applying the torch. Some of the replacement 62 series were diecast.
Jon.