2002 Volkswagen GTI won't go into gear when car is on.

If all the gears engage when it's not running then I could only assume the linkage is fine.

It could be that the clutch is worn out or the hydraulic system that operates it is bad.

The clutch is less than a year old and their was no problems we could find. So we gave in and brought it to the vw dealerships. Turns out the clutch got wet and stuck to the flywheel.

I had the identical problem with my 2002 GTI manual trans. While the engine was running it was not possible to shift out of neutral, Turn off the engine and shifting worked fine. Well there was flooding in my area the night before and I had to drive through some deep water.
This post really helped me identify my problem. Unfortunately there was no suggested fix.
In my case, the next evening I let the engine idle to warm up and then pushed in the clutch and revved the engine a few times, Fixed-shifting back to normal.
Even my local VW mechanic was not aware of this
.
I'll think twice before driving through deep water again!