O3 saturn vue power steering works when it wants!!!

Your steering wheel might be in a bind with the key lock. Try turning on the wheel while you turn the key.

Jump the battery. If it gets to low, the solenoid for the key won't work and release the key. Just a note. If it is that low replace the battery. Have it checked first, but most likely dead.

Have you checked the power steering fluid?

Did you disconnect the PS lines of the rack and pinion? Raise the front tires off the ground and fill the PS reservoir. Turn the steering wheel at least 10 times left and right till it stops(purge the air). Check the reservoir, fill as necessary. Do this twice. Then start the car and turn it again back and forth. Seems fine, check the reservoir again. Take a road test. Good, done. Not good let me know.

Check the fluid level belt could be slipping might need a new one

There is a TSB for your complaint from Saturn: Here's the jist of it.

Condition: Some customers may comment on a squeak or squeal noise coming from the base of the steering column while turning vehicle.
Cause: The squeak or squeal noise may be caused by insufficient lubrication of the intermediate steering shaft-to rubberboot- contact area.
you may be able to get it warrantied through saturn. this is the TSB #
#03-02-34-002

i am a saturn tech and ive never heard of this concern,the columns do go bad but they just set a ses light and cut out power,the column could be bad,it could also be the intermediate shaft u joint binding,,,i rec take to dealer, could be serious

mine did the same thing. Where the steering column goes through the firewall (floor) there is a rubber seal. That seal has gotten dry. Apply a small amount of lube where the shaft goes through and turn the steering wheel. This helped eliminate my squeal.

it is more than likley the power steer pump not pumping or the belt is broken

Sounds like the fuel pump may be going out. By the way the power steering is electric in Saturn, so it does not require the engine to be running.