The Toyota dealer put in a new battery , 2004 Sienna, I don't know the history. Since the new battery, the car will not crank, no solenoid click, headlights and all accessories work fine. When jumped, it starts immediately, you hardly hear the starter. I have loosened and re-tightened the battery connectors and secondary cables at the terminals, everything looks good. We will be going back to the dealer soon but I would like to hear some ideas , in case he tries to sell a new starter, etc. New info --- After several jump starts , now it won't jump start - no solenoid sound. I suspect the battery has nothing to do with it. Ignition switch ? Transmission interlock switch?
For a small fortune ,the dealer said it was the starter. Because the starter turned just fine when it turned, I believe he meant the solenoid. And the facts of life are a dealer is going to replace the starter not just the solenoid. I guess the days are gone when I cleaned up the solenoid contactor for free and drove another 100 K ; maybe spent a few dollars on starter brushes.
Firstly check the battery connections are nice and tight with no corrosion on the terminals or wires.
Look at the connections on the starter itself and make sure they are nice and tight with no corrosion.
Seen as you hear no clicking I would assume it's the starter solenoid that is at fault.
Hope this helps.