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I can't tell from those symptoms what is the problem. Some testing is in order, IMO.
First thing I would check is fuel pressure. Put a gauge on the Schrader valve and tape the gauge to the windshield.
(be sure the gauge fitting is firmly and well attached to the Schrader...don't ask why I caution this...)
-turn on the key and see of there is pressure and if it holds.
-start the engine and record fuel pressure.
-blip the throttle to see if pressure holds (does not drop).
-when it fails to start check the value of fuel pressure.
These will tell you if the problem is in fuel delivery or not.
Could be a lot of things. maybe a vacuum line off, maybe the VATS, maybe bad components in the distributor, maybe fuel supply (which has it's own list of maybes) as stated above.
How long does it run before it stalls?
Are there any trouble codes captured by the computer?
Does it have spark, fuel pressure?? Will it not start unless you reset the computer?
I agree that spark and fuel pressure being the first things I'd check if it was my car. I'm in no way an expert but additional things I'd do if it was my car I'd clean up all the grounds in the car. I don't believe VATs is part of your problem as VATs shouldn't come into play except in starting the car, once started it's out of the picture. I don't have my manual at hand but I think there's a ground strap and some other contacts in the distributor (some which I believe require dialectic grease) that should also be checked.
Basically it sounds to me like something electrical is opening with heat during operation and those areas should be checked first.
I don't believe VATs is part of your problem as VATs shouldn't come into play except in starting the car, once started it's out of the picture.
Friend of mine had an 88 this was a long time ago and it would randomly shut off the car while he was driving. I was not there but the story I was told was it was the VATS losing connection with the chip.