Had to jump the car after it sat a few days. Took it to a local shop for a starting/charging system diagnostic. They indicated it needed a new battery and alternator. I have a new battery and will take it in to have the alternator replaced - but now I can't start the car with a jump. I get lights, then one click, then nothing. Same as before.

Put the new battery in, same no start.

Wouldn't the car start if just the alternator is bad?

 

I think your starter is bad. When mine was bad, the car clicked once and I could hear a very faint whirring/buzzing sound. I just re-built my starter and it worked fine.

 

I hear one click when I turn the key. Is that the solenoid? Or does the solenoid produce a series of clicks?

 

Sound like you got a bad starter. If this happened again, try to tap the starter by using a long piece of wood and hammer it on the starter and then try to start the car up again. The 99 ES is easy enough to get to the starter.

 

Is the starter right next to the battery and air filter housing? Does it say 'Denso 12V' and '1F'?

 

Yes, and it's mounted of top of transmission.