My car won't start wHow to reboot

Can you ensure the car is in park?
Move lever to D and then back into Park and see what happens.
I have heard with this particular model, we have the environmentally unfriendly RX 330, that the batteries lack of charge may play havock with the electrics in these cars.
I religously charge my battery every six months with a smart charger so its always in peak, I also find a quick zap at the first cold snap of autumn / winter gives us trouble free motoring.

All fuses good? If all fuses good take the battery cable off and let it reboot for about 15 minutes than hook it

You will need the dealer to look at his, diagnosis of this problem takes special equipment and skills.

Are you saying the engine cranks but won't start, or the engine won't crank ?
If the engine does not crank, you would check for voltage at the starter, the neutral safety switch, and the key switch as well as the ground connections.

call your dealer

Maybe your on board computer is having a little bit of a hick up. Try disconnecting your negative battery cable for about 5 minutes and then reconnect it. This will totally reboot your computer. Hopefully this solves your problem. Best wishes and good luck.

check the fuel pump fuse in the fuse box

If you lost the only key you had, then go to the Lexus dealer. You have to show them a picture ID, and the car ownership (TITLE). Is going to cost you, so it is a good idea to call them upfront.
Another option is buying a key blank transponder chip on line at:
http://www.autotransponder.com/2003_lexus_lx470_key_blank_transponder_chip_2

2003 LEXUS LX470 Key blank (Transponder Key) must be cut and programmed before it will work for your vehicle. Cutting and programming can be performed at an automotive locksmith shop or your local dealership. There are no onboard programming instructions available for 2003 LEXUS LX470 chip key. Only a professional with special tools can make it work for your vehicle.
Or at:
https://www.ikeyless.com/vehicles/Lexus/LX-470/2003