Toyota 2004 Highlander- Headlights work but no high beams. Hi beam relay contacts close and put out power. Seems like no power into wiring harness. I can split the fuse box at the horizontal seam. Is it ok to disconnect the power supply wire and open it up to check connections at relay terminals? Is there enough slack in the wires? I don't want to pull all the wires out. Will it power off the computer? Is this ok? Running lights don't work either.

The relays are next to your battery. The first one is the horn relay (the yellow one closest to the headlight), then the lo and high beams after that, I forgot which order, and then the starter relay.

Get a freind to check which one clicks when you turn on the high beams (put your finger on it), and swap it over with the one next to it.

Check to see if the headlights work, but the high beams not. If this is the case, the relay is fried. If the high beams are still not working, then check the contacts in the combination switch (on the steering column), or the wiring... this is a bit more difficult to trace, none the less, not that difficult.

This one is pretty easy to conceptualize if you understand how a relay works - and if can't then you should go read my All About Relays page so you can. You splice two 20A relays into the existing wiring harness right out near the headlights so one relay controls the low beams and one controls the high beams. Use the existing high and low beam wires coming from the firewall to trigger the relay, run a new high power feed (with a fuse!) direct from the battery, and hookup the existing high and low beam wires from the headlights to the "normally open" contact on the relays. The hardest part of all this is typically finding the right wires in the existing wiring harness and finding a place to mount the relays - the actual wiring is pretty easy. Any SPST/normally open relay will do, though most automotive relays are of the SPDT variety - just don't hook anything up to the "normally closed" contact on the relay (pretend it's not there) and you'll be fine.
Note that if you pick some really monster sized off-road headlights that draw more power than the ones that you can plausibly use "on the street", you must use higher amperage relays than what is mentioned here. You must also use the appropriate sized power wires and you may very well end up replacing all of the headlight wiring from the relays out to the headlights themselves

High beam switch moat likely, have you checked relays and fuses?

Try the bulb itself. You can check it with a voltmeter set to continuity. Check the harness for a poor ground connection near the bulb socket. It will be a black wire under a screw head on the chassis. If nothing out of the ordinary is found, replace the headlight relay for that side. Each headlight has its own relay.

If you have the HID bulbs check with an Autoparts store like Autozone. I believe the right and left headlights have separate Relays in the Powerbox. It is possible that there are 2 low beam and 2 high beam relays total. They pull too much power for both sides to be wired together and may wire the high beams separate too!

Power for the headlights on the Left side (High and low beam.) comes from the same fuse... The ORANGE wire and the ORANGE/BLACK are the same, they connect in the harness. If you only lack the 12v to make the light work connect them at the headlight assembly.

If your low beam also lacks ground the Left low beam ground is shared with the right low beam ground... Run a wire from left low beam DARK BLUE to right low beam DARK BLUE/WHITE.

If its not working now then your cursed, seek a witch doctor.

it sounds like a bad light switch

Power flows from the battery to the main fuse. Then it flows to the control coil and contact side of the headlight relay. When you turn on the headlights, the headlight switch provides ground to the control coil in the headlight relay and that operates the contacts. The contacts allow power to flow to the right and left hand headlight fuses. Power flows to the headlights. The headlight switch is what provides ground to the high and low beams on the headlights.

To diagnose, remove the headlight relay. Using a voltmeter, check for battery voltage on terminals 3 & 4 in the socket with the key in the RUN position. Then turn on the low beams. Test for good ground on terminal 1 in the socket. If you're not getting ground, then you've got a problem with either a bad headlight combo switch or a bad ground. If you get good ground on terminal 1, reinstall the relay, turn on the low beams and test for power in the headlight sockets. If you get battery voltage, that confirms you've got good fuses and wiring up to that point. Next, check for good low beam ground on the red/green wire in the bulb socket. If you're not getting good ground, it's either a bad combo switch or bad ground from the combo switch. Then turn on the high beams and check for good ground on the red/yellow wire in the bulb sockets.

It wouldn't hurt to clean the ID ground on the left side of the instrument panel, but based on everything you've said so far, this sounds like a bad combo switch

check bulbs first. then fuse and relay. can check for voltage to headlight with high beam selected. If none of the above it just about has to be in the switch

The only thing that all of your problems have in common is the light switch and the dash wiring harness. The stereo is linked to the light switch because of it's illumination feature' You need to test the light switch with a test light by probing the wires at it's plug.