For safety reasons VATS won't cut power to the engine while it is running. But as was said, it will shut down the injectors and prevent the start relay from operating the starter solenoid. The fuel pump is shut down automatically by the PCM if the engine is cranked for more than 5 seconds with no start. If you were to attempt to start the engine by shorting the solenoid directly to battery, the starter will engage, but the PCM will still shut down the injectors, and will shut down the fuel pump within 5 seconds to prevent flooding and washing oil off the cylinder walls.

Look for bad connections, an alternator on the way out, or possibly a battery that shorts out or goes open internally when bounced around.