'87 won't start when hot
 

Our '87 SR5 (224K miles) won't start if the engine is very hot. Always starts when cold/cool, and does not overheat when running. If parked for just a few minutes--like filling with gas--it will start. But, if parked for a longer period (5-10 minutes), and the engine gets real hot while sitting, then all I get is a click noise from the area of the starter. I'm thinking I need a new starter or solenoid, but would rather not shotgun this problem if someone else has found the solution.

So, do I need a new solenoid (a.k.a. Magnetic Switch) or a new starter, or both? Any help greatly appreciated. TIA.

you need to do some troubleshooting, put a meter on the positive cable at the starter and check your voltage, compare that to what you read across the battery. if there is any significant difference clean the cable ends and test resistance from one end to the other with it disconnected. Check all of your ground wires especially from the battery to the block.

"The area of the starter" will not help people who are trying to help from thousands of miles away.
Pls describe in exact terms what you see, hear, feel, smell about the issue.
Are you absolutely, positively sure and swear the click is in the starter area and not behind the glove compartment?
If in the starter area, exactly what clicks? If starter solenoid, does it click energetically? If it clicks energetically but you don't get a crank, yes 99% sure you need new solenoid contacts.

Whether you need new solenoid contacts or not, I'm still absolutely, positively sure that Toyota wired your starter relay wrong at the factory and even documented it (here).

Even if you fix the no-crank problem and mask the wiring problem with new solenoid contact or smoothly operating starter solenoid, the wiring problem still exists.
Here's how to fix it for about $11.