2004 chevy trail blazer, frt. blower motor runs with engine off, ignition key removed from switch.

First test the blower motor by jumping 12 volts to the motor connector. If the blower runs pull the dash switch and check for a burnt connector and overheated wiring at the blower speed control switch, if that is the problem, replace the switch and the wiring harness to the blower motor. I think there was possibly a Recall or Technical Service letter on this problem.

Note the diagram for blower motor. Note the fuses, not only check the fuses, check the fuse circuit for voltage, use a test lite. Some of those fuses get ignition feed, they go hot with the key on.
You may have to check feed to and from ignition switch when problem is ongoing? Could be problem with ignition switch? Don't replace anything unless your testing points in that direction.
At the blower motor wire connector, purple wire is voltage, black is ground.

Possible ignition switch failure. Try gently rocking the key back and forth slightly with engine running to see if they begin to work again. If they do, suspect an ignition switch failure

Hello Linda! Replacing the HDM relay for the low beams will not trigger any of these problems. If the blower isn't working, you probably need to get a new blower motor resistor (for manual controls) or blower motor module (looks the same) for automatic climate control. To access, need to remove the plastic kick panel near your feet on passenger side. After removed, look up you'll see the resistor, it's about 4" long with 2 small screws in it. You'll also need to drop the glovebox door to remove the connector from the resistor to the motor. Suggestion- buy a pig tail for the resistor also, these tend to melt. Very common problem. The ABS dash could be a bad wheel hub assembly or a hub starting to go bad.

Faulty blower motor resistor or blower motor resistor connector faulty.Wiggle blower motor resistor connector with the ignition and fan switches on.

The most likely culprit is a bad blower motor resistor.

If you have too many keys on your keychain, the extra weight can wear out the key tumblers in the ignition switch. If so, you will need to replace the ignition switch key assembly with a new one.

This vehicle is a chevy trail blazer , doesn't matter yours is in the same place an the samething .
Chevy Trailblazer EXT Blower Motor Won Turn Off

If you have the automatic climate control and the blower motor won't shut off then you have a failing Blower motor speed control module. Something inside this module goes bad and the result is a blower motor that keeps running at times even though the engine is off and the key is out of the ignition. The module isn't repairable but has to be replaced. I believe it is located behind the passenger side of the dash panel.

ignition switch is bad ... you should NOT be able to remove the key while running (safety feature on all US cars since 1970's).     Brake switch is hooked into ignition then to shift lever to prevent moving from park unless ke is on and brake is depressed ... your switch (or wiring, but probably switch) is shorted out and/or maybe has mechanical damage (from when the thief tryed to start it with a screwdrive?)