No turn over could be due to several things.
1) PATS: is the yellow lock flashing on the dash when it doesn't crank?
2) Ignition switch
3) Neutral Safety switch: try shifting out of park into N and see if it will crank.
4) Starter relay: this one you'll have to simply buy, I don't think it will swap out with another relay- nothing else is the same size.
A relay might seem to be a simple shot, but I have only ever known a handful of bad relays in 25 years of fiddling with cars. It can happen, but really, check other things. I'd be more certain that the ignition switch was the problem, but I can't explain why it would need to cool off. Likewise with the N/S switch which is in the shifter and prevents you from starting the car in anything except P or N. I don't think this can be replaced without replacing the shifter.
A loose starter terminal or battery terminal can cause a no-crank issue when the battery is warm, but I'd expect some clicking.
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