My car won't start when it's hot outside

So you are saying if the interior is hot, it won't start (no crank, not even a click), but if you roll down a window and cool the interior down it does start? First, check you don't have the anti-theft system activated when this occurs. If you have any aftermarket items that interact with the anti-theft, like remote start, heat may be affecting wiring where those systems were spliced in. Second, heat may be soaking the starter or causing a intermittent component failure in the starting circuit. To isolate that it would need troubleshooting. (A bench test of the starter would confirm its operation.)

Finally, I run across someone who is having the same issue that I am having. I also have a 2002 Toyota Avalon and I'm having the same problem. If it has been a hot day, windows rolled up and the humidity on the interior of the car is unbearable, I know before I put the key in the ignition that it won't start until I cool off the inside of the car. The fix is to just open my door and just wait, sometimes 10 minutes then viola it starts. This problem seems to only happen in the Spring and especially the summer and no matter how many mechanics that I mentioned this to through the years, they have no idea. I'd hate to get rid of the car because it's paid for but I find it odd that nobody has a solution for this issue.