1997 Camry Fi Problem

Hey all, I need your help. I've got a problem with my sister's '97 4cyl toyota camry. I've narrowed it down to something electronic. Here are the symptoms and my reasoning.

Car runs totally fine when its moving. If it is stopped and idling, it can idle for maybe 5-10min then it dies. I had the hood open looking at it and inspecting it for a good while and left it idling while i checked a noise id been hearing coming from the suspension. Idled for maybe half an hour like that but once i shut the hood it started running rough and shortly thereafter it died. We then proceeded to drive it home almost 500 miles up I-5 that night with no issue. When it dies and we try to start it, i do not smell fuel out of the exhaust so im assuming it is fuel related. Once its left to cool for a few mins it will start back up and run fine except the check engine light comes on until it gets moving and fresh air gets through it.

My thoughts are that when the ambient temp. in the engine compartment gets too high that it is causing one of the electrical components to act up (such as a resistor inside a circuit or something similar). Once it cools, the check engine light goes off and all returns to normal. We've already replaced the fuel filter thinking it was clogged and the fuel pump thinking maybe it was going bad but that was before I really got a chance to look at it and drive it.

So the check engine light really does not stay on long enough to check it (i did buy an OBD2 reader on ebay and plan to check it but it wont be here until 2 weeks from now). Any thoughts?