any idea why a 2003 toyota tundra won't reverse?

First, I'd call a dealer or go in person with the VIN and find out if there was a transmission recall for the y m m. if none, I'd find out more about the transmission and it's deficiencies, etc. The less money I put into it, the better off I am. It sounds like the reverse band is stripped / worn out OR you have a diff problem on the back axle. If it's not grabbing, it's not going anywhere. If it's not been maintained properly ( fluids and gen main) it's going to give you fits. I've never heard of a transmission only working when it's warm ( especially in the summer). Release your e brake