I have an 07 Vue recently purchased that runs and drives great in every respect except one. Every time I place the car in reverse the idle drops and the motor shutters and cuts off. There is always an odor that smells like gasoline or the smell that associated with a car flooding. There are no trouble codes. I have checked for vacuum leaks and there are none. This does not happen in drive and once the car is moving forward it runs great. I have taken the car to several shops including the GM dealer and no answers. I have received suggestions from replacing the ECM, replacing the throttle body, and even replacing the engine and transmission. Does anyone have any suggestions?

Smell of gas: Could it be related to the fuel pressure regulator? https://www.700r4transmissionhq.com/...ure-regulator/
The IAC -intake air control- controls the idle. If it is stumbling or wavering, I would get a new one.
A higher line pressure may be required when placed in reverse especially when fluid has not had a chance to fully warm up.
Is the trans fluid at the correct level when hot? On hoist, a shop should be able to tell you if the level is correct.
Where is the idle speed at before you place it into reverse?
Have you had the transmission fluid & filer serviced recently?
If not, consider getting it done and use the Dexron VI (6) ATF fluid.

Put in reverse the engine torques opposite. Have someone work it while you look and see if it pulls an any wiring, plumbing etc.

Idle speed is around 1000 RPM when put in gear. When I put the car in drive it initially drops to around 800 for a second then recovers to 1000RPM. When I put in reverse it drops to around 800 RPM but never recovers and eventually dies out.

This is where I want to know if GM has any technical service bulletins about this stalling in reverse issue re your 2007 Saturn Vue.
I'd like you to call GM/ Saturn at 1-800-553-6000. Hopefully this is the right number for customer care.
Tell them your story, & what GM dealer you were at, that seemed to be un-helpful.
Since you are not the only one with this problem, possibly GM does have a tech bulletin in their files.

What engine does it have?

The car has a 2.2 liter or 4 cylinder engine. I checked with GM and there are no bulletins related to this specific issue. The car has just started intermittently doing the same thing in drive. It does not cut off when I initially put the car in drive but when I stop it will stutter and eventually cut off. The same smell of fuel is present..

"Soft Ride" mentioned that it may be the fuel pressure regulator. Do you have a part number for that specific part. The part store tries to sell me a MAP sensor which I have already replaced, with no effect.

Someone mentioned fuel injectors suggesting one may be weak. Could this be a possibility

I failed to mention that I have already replaced the throttle body, ECM, and one mechanic "cleaned" the converters. No change

Called a GM dealer for you.
They call it a fuel pressure dampener. Part number is 12596245 for the 2.2 engine, & it'* compete with a new rail as far as I know.
Due to the smell of gas, If it is not leaking fuel anywhere , that probably is not the issue? As the car overall runs fine, expect stalling in reverse.
Is the rail wet or smell of gas present along the rail anywhere?
Have you checked the transmission fluid level yet??
This may be a transmission related reverse issue also Low line pressure, stuck solenoid, valve body, just a thought.
.

Not sure what they did to "clean" the converters, but that'* usually snake oil and/or damaging in my book.

How many miles are on it?

How many miles have you owned it?

How many miles on the spark plugs?

What is the fuel pressure when this happens?