Sorry for the delay I typed out a long explanation only to have the program crash.
A primer bulb takes the place of a fuel pump to fill the engine with fuel. As you squeeze the primer bulb the carburetors or VST in the case of fuel injection has an inlet that lets the fuel in. Once full the inlet closes and the bulb gets firm. Once the engine starts running the inlet opens to allow more fuel to enter and the bulb softens. As long as the bulb doesn't deform or flatten you should be good to go. When running the bulb is just a part of the fuel delivery system. Being the softest part of the system if there is a restriction it flattens first.
You can rule out the fuel delivery by running the engine on a portable tank and different fuel line.
Being that you have a HPDI engine some of the fuel pressures, particularly while running or shortly afterwards, are extremely high. It is an engine I don't recommend someone without the training to work on without the training or proper tools. So a trip to the local shop would be in order.
Now that doesn't mean you can't check some things.
Start the engine. While running slowly have a helper squeeze the primer bulb. Does the engine stay running? If so then the lift pump, a low pressure pump that brings fuel to the engine may be due for an overhaul.
Try this and let me know what you find. We'll go from there.
Finally do you have any fuel injection tools or background?