Accessories (windshield wipers, radio, etc.) not working when car is running However, when I trun the ignition switch to accessory everything works? What is the problem?

First you need a wiring diagram, free at http://www.bbbind.com/free_tsb.html Then you need a DVOM - digital volt ohm meter to test the circuit . Then if your not familiar with automotive electrical systems testing ,go on youtube an check out Electric Testing Techniques You Need to Know Also having a factory diagnostic trouble shooting chart would also be usefull . You could just start replacing parts , you mite get lucky . The wiper motor is a high fail part , so is the switch ..You could find the fuse is blown ! But fuse's don't usually blow without a reason . Short to ground , low resistance . ( high current flow) Also knowing how they work and what all is involved in making them work.
The Wiper/Washer System consists of the following components:
• Windshield wiper/washer switch
• Windshield wiper motor module
• Windshield wiper motor
• Windshield washer pump motor
• FRT/WPR/WSHR 25A fuse

Wiper/Washer System Operation
The windshield wiper motor module is part of the windshield wiper motor cover and controls wiper motor operation. The windshield wiper motor is a 2 speed motor and is operated at low speed in all modes except HIGH. The accessory voltage supply circuit to the windshield wiper motor assembly is used to operate the wiper motor in all of the low speed modes. The accessory voltage supply circuit to the windshield wiper/washer switch is used to supply the windshield wiper switch signal circuits and the wiper motor high speed circuit.
When the wiper/washer switch is in the LOW position voltage is supplied through a 24K ohm resistor within the switch to the windshield wiper switch signal 2 circuit, and accessory voltage is supplied to the windshield wiper switch signal 1 circuit. The reduced voltage from the wiper switch signal 2 circuit and accessory voltage from the wiper switch signal 1 circuit causes the wiper motor module to close the wiper motor accessory voltage supply circuit to the wiper motor low speed terminal.
Windshield wiper/washer system MIST operation is identical to LOW operation except that the MIST switch is a press and release type. When the wiper switch is moved to the MIST position and released, low speed wiper motor operation is started and will continue until 1 cycle is complete. If the wiper switch is moved to the MIST position and held, the windshield wiper motor will operate in the LOW mode until the switch is released.

Don't the wipers work in any mode ?
1
Did you review the Wiper/Washer System description and operation and perform the necessary inspections?
YES - Go to Step 2
NO - Go to Symptoms - Wiper/Washer Systems
2
Turn ON the ignition , with the engine OFF.
Operate the windshield wiper/washer switch through all the switch positions.
Does the windshield wiper/washer system operate normally?
YES - Go to Testing for Intermittent Conditions and Poor Connections in Wiring Systems
NO - Go to Step 3
3
Turn OFF the ignition.
Disconnect the harness connector of the windshield wiper motor.
Turn the ignition ON, with the engine OFF.
Connect a test lamp from the accessory voltage circuit of the windshield wiper motor to a good ground.
Does the test lamp illuminate?
YES - Go to Step 4
NO - Go to Step 13
4
Connect a test lamp from the accessory voltage circuit to the ground circuit of the windshield wiper motor.
Does the test lamp illuminate?
YES - Go to Step 5
NO - Go to Step 14
5
Connect a test lamp from the signal 2 circuit of the windshield wiper switch to a good ground.
Press the windshield washer switch.
Does the test lamp illuminate?
YES - Go to Step 11
NO - Go to Step 6
Let me know if you make it this far and i will send rest of diagnostic's

Does the heater motor work? Possible ignition switch causing problem. Turn wipers on and gently move key back/forward slightly with engine running to see if these accessories start working. If they do, ignition switch is at fault

wipers should normally work in the run position of the ignition switch. Accessories position is for radios , cig lighter and other items that you need when stationary with the engine off.

Those are accessory functions working off the ignition switch when in accessory position or run position of the key. Were you to check if the wiper and radio fuse had power to them in the run or accessory position as they should, likely they would not. Power from the switch goes to the fuse panel, powering the fuses and circuits that only work with key in on or acc., either a direct wire to the fuse panel or sometimes, fuses get power from an accessory relay that sends power to fuses, but relay has to get a signal from a good working switch before it powers the fuses.
What I'm trying to say, is the bad switch is probably the reason you have no wipers, radio, or mirrors.

I am betting it is your ignition switch- it is just like any other type of switch, in that it has contacts that eventually wear out. Once you turn the key to start the car, it returns to a position that maintains the contacts to power your accessories, and it sounds like those contacts inside the ignition switch are worn or broken. The "acc" contacts when you turn the key backward are rarely used so they are still in good shape, unlike the "run" contacts which get hit every time you start the car.

Eg wipers work while ignition switched to 'accessories' but stop when key turned to 'start'

I had this issue and took me ages and ages of wire tracing and probing. Your fuse box should have the wiper/windows accessories fuses. If you probe them and they aren't showing voltage, check wiring harness, check wiper motor voltage. My ignition barrel was the fault of all this, everything worked when switched to "accessories", but while the engine was running, the accessories turned off. The bottom of the ignition barrel pushed out, as the factory stopper groove bent out. All I did was cold chisel a dent back into the case. BAM fixed my windows/starting/wipers all in one go.

Could be anything but based on experience the probability points to the self cancelling switching in the motor gearbox.

thats a possibility!

Change the ignition switch

I think a new ignition switch would avoid the torching of your sick van.
It may not help but try this:
- Make sure you have the radio on in the accessory position, then switch to the 'run' position; you needn't start the car.
Jiggle with (deserved) anger on the key - well, without breaking it, and watch the radio display to see if it will flicker when you do this. If it does, then the switch section that should contact in both Accessories & Run has failed. 
Most radios don't instantly turn on but delay a second or two,  therefore the attention to the display.