Just bought a 2002 ls430. The interior lights don't work; and none of the accessory functions such as power locks, windows and wipers operate unless the engine is running.

 

that is an interesting problem. i suggest disconnect the battery, and pull out and test every single fuse and relay from your electrical box as a start. you can buy a relay tester on amazon

 

Sounds like a duff battery to me. Since the LS430 uses a CANBUS, low battery voltage can cause all kinds of weird effects.
Once you start the engine the alternator will be supplying 13.8volts so the CANBUS will work as it should.
Fit a new battery even if yours seems to be OK and starts the engine.
If it's not that, then there is a problem with some of the relays.