2003 Camry 4cyl won't turn over when driven long enough to warm up

Seems to indicate a problem with the starter. The starter draws much more power to crank the engine than minimally powered devices such as lights, accessories, etc.. Normal operating temperature heat affects a starter adversely in a weakened condition, or sometimes it can be a condition known as 'heat soak'. Heat soak is a condition where a heat source such as the exhaust is too close to the starter causing an overheated condition, explaining why the car starts again after it cools down. More likely though, if there have been no changes to the exhaust, including leaks that could blow directly on the starter, it would be a weak starter. My bet is you need a starter.

I have the exact same problem!!!

BSling's answer cured my problem. A new starter was the answer, and the car is fine now.