Take the inlet fuel line off the pump. Hold your finger over the inlet hole in the pump while someone cranks the motor to see if you feel a good vacuum there. If you do not, the pump has a valve problem. Do you have a filter before the pump? On older vehicles like this it is a good idea because the fuel tanks are usually not clean. The filter catches particles before they can get into the pump valves and cause them to not seal properly. If the pump body has screws around the periphery, you can take the pump apart and clean the valves and any other crud out of it that might be present. As a precaution, I would also take an air hose and blow back through the tank to pump line to clear any crud there that might be impeding fuel flow somewhat.
As to the comment that even if the pump valves are letting fuel drain back to tank there should be enough in the carb to start it after sitting a week or so, I find that if mine has run long enough to get up to temp, when shut off the fuel can get percolated out of the carb and also can evaporate over time. It would be a really good idea to source a proper pump with a priming lever. I find I have to prime mine at times because it invariably will sit for longer than the week you mention!
Regards Marti