Suspect a problem with the crankshaft position sensor. Pulses from it tells the engine computer the engine is rotating. At that time, the computer turns on the Automatic Shutdown (ASD) relay which sends voltage to the coil(s), injector(s), fuel pump or pump relay, alternator field, and oxygen sensor heaters.
Also feel if the ASD relay clicks for two seconds when you turn the ignition switch on. If it doesn't, the terminals or coil inside could be corroded.