2002 Camry cabin fans not working in hot weather

I have a 2002 Toyota Camry. I live in Tucson, AZ
where the temperature outside reaches well over
100° on a regular basis. I am at an apartment
complex with limited covered parking. Whenever I
don't get a covered spot and have to park in the
sun, my fans won't turn on when I go out to my car.
They usually turn back on after 5-20 minutes of
driving. Yesterday was different: I was parked in the
shade and they wouldn't turn on and they still
wouldn't turn on after over 30 minutes of driving.

The a/c itself is producing cold air (I know because I
turned off circulating air so outside air would come
through the vents when I'm moving and I feel cool
air being pushed out). My engine is not overheating
and I know the coolant system is fine because I've
had it checked multiple times recently. I have
checked the motor plug under the glove
compartment and it is secure.

Do you have any idea what's wrong? Is there any
way I can fix this myself? Or do I need to take it in?