Could be a few things.
EEC relay powers the computer(EEC), fuel injectors, and spark.
So if it is not staying closed after startup system would die, just like key was shut off.
Fuel pump relay, same issue, you should hear the fuel pump come on with the key then go off 2 seconds later, it won't come on again until engine rpms are above 500, so after startup, if relay isn't closing again engine will die within a few seconds.
Ignition switch, under steering column, it has a START and a RUN position, when you turn the key to START power connections change, when key returns to RUN contacts change again, because RUN is the normal contacts they could be worn down so power is being lost in RUN, EEC relay opens and engine dies.
Do lights on the dash go off like key is off or do they stay on like key is on and engine is off.
TFI ignition issue, older they get the more likely they will have a problem.
Good read here for testing:
http://www.therangerstation.com/tech_library/TFI_Diagnostic.shtml
Just to narrow things down, you could get some Starting Fluid spray(ether), then remove intake's air plenum(big air tube) have someone start engine and when it starts to die spray some starting fluid into the intake.
If engine stays running then fuel supply is being lost, but spark is working
If engine still dies then spark is being lost
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