Wont start manual transmission can start by manually overriding the starter relay

That vehicle does not have a starter relay. Does the vehicle have automatic or manual transmission? Turning the key to start sends battery voltage to the clutch switch with a manual trans. and trans. range switch with automatic trans to the s terminal on the starter solenoid. You probably have contact problems inside ignition switch.

You can have a problem with the shifter cable. It will only allow it to crank and start if it's in Park or Neutral and it may not think it is. Move shifter through its gears one at a time and see if it tries to crank. That is assuming it's an automatic transmission. If it's a manual it could be a clutch pedal switch. Could even be a starter relay or loose wiring between relay and starter. Good luck.

I could not find one on my 1988 Camry either but if I read the shop manual correctly, there is no starter relay for cars with an automatic transmission. Cars with a manual transmission use a starter relay. There are other obvious errors in the shop manual wiring diagrams but this is what it implies.

does the clutch push normally or is the linkage broken not disengaging clutch, and then does shifter move around or locked and wont move, loose linkage in shifter or gears are jammed inside transmission and something went, if shifter wont move at all then trans is jammed inside and needs repair.

Is this a automatic or a manual transmission if it is a manual transmission the clutch pedal switch is non op. If this is a automatic transmission your concern should start at the starter motor relay this is a fused relay that is operated by the ignition switch. Make sure you are dealing with a faulty anti theft unit first.

Since you have power to your accessories, here are a few more things to consider:
1: Automatic transmission not in park or the neutral safety switch is faulty
2: On manual transmissions clutch pedal not depressed or or clutch start switch failure
3: Broken or loose wires in the starting circuit. Inspect all wires and connectors at the battery, starter solenoid, and ignition switch
4: Starter motor pinion jammed in flywheel ring gear. Manual transmission. place transmission in gear and rock the vehicle to manually turn the engine. Remove starter and inspect pinion and flywheel

Are you using a screwdriver because the ignition will not cause the starter motor to rotate? If so, there are a few reasons that can be:
1) The starter fuse is blown
2) The starter relay is defective
3) The clutch start switch is defective, or the wiring harness is not making a good connection ( in case you have a manual transmission). A similar switch is mounted on the automatic transmission to make sure the car is in park or neutral when starting.
4) There is also a 60 and 100 amp fuse between the battery and the ignition switch that could be defective.
5) The ignition switch is defective.

When you do use the screwdriver, are you putting the key in the "ON" position? This should start the car if the starter motor is in fact rotating.

the neutral switch is located in the left side of the transmission and its connected to the shift cable.

does the starter solenoid get power when the key is held to crank and the clutch pedal depressed?

if it is automatic transmission try to start it in neutral, and see what happend, but if it come whit manual transmision, then check the cluth switch, you can by pass the switch and try to start it, if it works then replace the sw, but if not the look to see if there is a starter relay and replace it, most of this come wiht starter relay and they fail.