When you pump the bulb you are acting as the fuel pump.
Now, it could be that the tank vent is blocked and the fuel pump just didn't have enough "suck" (and your pumping the bulb forced some gas up) OR your fuel pump might be on it's last legs.
If you have a built in tank you will have to search for the vent and then try and see if it's clogged.
If you are running on a portable tank you can 1) make sure the vent cap is open or 2) totally remove the cap and try running to see it that works (just making sure the tank can breathe).
There also could be a clog in fuel line between the tank and the motor, at any of the connectors or in the fuel filter(s) if present - again, pumping the bulb may be forcing enough gas past a clog to allow it to run.
If the tank/vent is "ok" and nothing else appears to be clogged, then you need to have your fuel pump inspected and possibly rebuilt.