My 01 camry has recent intermittent starting issues.. when you turn the key to start the car, you can hear the "click" of the electrical engaging but the starter doesnt "engage"... if you jump start the car or "boost" it from a battery charger it does start... battery checks out fine.. doesn't do it all the time.. starts cold most days fine... during trips with multiple stops it may do it during one of the stops... ?????

 

Typically issues are battery cable connections, battery terminals attachment to the cable, chassis ground connection for negative battery cable, there is also a jumper cable between the chassis and engine block.

Or could be the electrical contacts inside the solenoid. The solenoid is bolted to the starter motor, follow the positive battery cable to the solenoid.

The contacts can be replaced with the solenoid/starter unit still on the car. The Toyota dealer sells them.