Steering vibrates when its in gear. - 2006 Pontiac Vibe

This could be a few things. Does the vibration happen only when your moving or when you've stopped as well. Is it above a set road speed as well as RPM? Vibrations can smooth out as road speed gets higher.
The first, and cheapest, thing to check is your wheels and tyres. Get them re-balanced as a missing tyre weight can cause vibration. Also get the Tracking and alignment checked. If one or both wheels aren't lined up properly vibration occurs and it wears out your tyres faster.
If those are correctly set, get all the joints in the suspension checked as a worn ball joint or bushing will cause juddering.
Finally get the brake system examined. Badly worn brake pads and warped discs can also cause wobble.

Sounds like bad engine/trans mounts. The more load you put on it the more vibration engine makes. If mounts are bad, vibration will transfer to body and steering

check all suspension and steering ball joints , also check all 4 drive shaft cv joints . any of the above can cause this fault.

broken engine mount

Hello! This is a known problem that is addressed by the following Technical Service Bulletin...Guru...saailer

Subject:Steering System Moan, Groan, Shudder and/or Vibration Type Noise During Low Speed Turning Maneuvers (Replace Steering Gear) #04-02-32-007A - (12/01/2004) Models:2004-2005 Buick Rainier
2003-2005 Chevrolet TrailBlazer, TrailBlazer EXT
2003-2005 GMC Envoy, Envoy XL
2004-2005 GMC Envoy XUV
2003-2004 Oldsmobile Bravada
Short Wheelbase Utilities Built Prior to VIN Breakpoint 52123375
Long Wheelbase Utilities Built Prior to VIN Breakpoint 56137280
This bulletin is being revised to remove the 2002 model year and add the long wheelbase utilities. Please discard Corporate Bulletin Number 04-02-32-007 (Section 02 -- Steering).

ConditionSome customers may comment on a moan, groan, shudder and/or vibration noise while making low speed turning maneuvers. Customers usually notice this condition in parking lot turning maneuvers.
Correction

1. Diagnose the power steering concern. Refer to Symptoms - Power Steering System (SI Document ID #670351).
2. Replace the steering gear with a newly designed steering gear. See the parts information below. Refer to Power Steering Gear Replacement (SI Document ID #742314).

Parts Information

Part Number

Description

Qty
88965356
Gear Kit, Steering
(Short Wheelbase Utilities)
1
88967181
Gear Kit, Steering
(Long Wheelbase Utilities)
1
Warranty InformationFor vehicles repaired under warranty, use:

Labor Operation

Description

Labor Time
E9740
Gear Assembly, Power Steering - Replace
Use published labor operation time
E2000
Toe, Front - Adjust

I’m happy to help further over the phone at https://www.6ya.com/expert/david_6df67de3b14de867

check engine and gear box mountings

A transmission problem would not make the steering wheel vibrate. Have you ever had the Transfer Case fluid changed? This is required every 50K miles. A qualified transmission technician should evaluate your problem.

IF GEAR SHIFT SHAKES THEN THE TRANSMISSION MOUNTS COULD BE BAD. HOPE THIS HELPS. GOOD LUCK

It is likely both your track bar and the lack of a steering gear box stabilizer. The bushings on your stock track bar can be replaced with polyeurethane or metal bushings which will help curb the problem.
The steering stabilizer attaches to your steering gear box and your frame allowing the steering to be more stable. I have put both on my truck and it drives like a caddy now as a result. If I had to choose between the two items, my money would go to the steering stabilizer first. I believe I bought mine here:
http://www.xtremediesel.com/bd-powerperformancedodgesteeringstabilizer.aspx
Hope this helps

Just guessing. Could it be a warped brake disk? That would cause vibration. And vibration in steering might conceivably cause splashing out the power steering reservoir. You would know if the disk were warped by a shudder when you brake at hight speeds.