Fuel Pumps
A liquid left to its own devices will sit still. The only way to move a liquid is to give it a push, which is usually done using a pump. A pump creates a pressure differential and liquids always choose to move the regions of lower pressure. (Don't we all?) The diagram at left shows the path the fuel takes from the fuel cell to the engine and then through the exhaust. The fuel pump is responsible for getting gas from the fuel cell to the engine.
Fuel pumps on NASCAR cars are mechanically driven (as opposed to electrically driven) for safety reasons. You don't want the fuel pump on a car that has crashed to continue to pump fuel.
There are two different approaches to fuel pumps. The default used to be that the fuel pump was mounted by the engine; however, there is an alternative approach, in which the pump is mounted nearer to the fuel cell. The latter was motivated by lessening the chances of vapor lock (Vapor lock occurs when fuel is vaporized by heat from other parts of the car. Fuel pumps are made for pumping liquids, not gases.) Some teams that use a fuel-cell-mounted fuel pump have an additional pressure regulating device closer to the engine.
The original fuel-cell-mounted pump was the Waterman fuel pump, which is run on the Chevy R07 engines, but I understand that the 2008 Toyota engines also use this type of pump. Waterman originally was the only one offering such a pump, but they are increasingly available from other manufacturers (like CV Products).
One argument in favor of mounting the fuel pump on the fuel cell is that it lessens the chances of vapor lock; however, if you have any type of pressure regulating device near the engine that is expecting liquid and gets vapor, you're still likely to have issues with fuel supply. Another positive of this type of fuel pump is that it should be more efficient in getting fuel out of the fuel cell, although I don't have any testimonials as to whether that is actually the case or not.
The big question, of course, is why this would affect some teams and not others. How much fuel is in the fuel tank, the type of fuel pump (some are mounted back by the tank and others are mounted by the engine), and even the way components are mounted relative to the effective net force (i.e. the vector sum of centripetal force and gravity) can impact whether or not the driver suffers from a fuel intake problem on the restarts.