What is an effective and natural way to remove soot and carbon buildup
from the glass of a wood burning stove?

I have a new wood burning stove, with only 3 fires to its experience
so far. After a single night of being left to smolder rather than
being extinguished, there is a buildup of soot on the glass of my wood
burning stove which "elbow-grease" alone will not remove. The
manufacturer and installer both recommend a commercial cleaner -
loaded with chemicals I can neither pronounce nor want sprayed within
my home... I am (somewhat ironically, I know, since this is a wood
burning stove) a tree hugger, and am not interested in any harsh
chemical solution.

Here is one suggestion:
http://www.worldwise.com/cleanstovand.html

The other sites seem all to recommend special cleansers that may be
what you are trying to avoid.

Hope it works.

No personal experience but a friend was discussing this over Christmas
and they recommended an old fashioned method of newspapers damped in
vinegar.
If this works I'll be happy to post it as an answer!
answerfinder-ga

I've found a wonderful natural method for cleaning the glass and it
works like a charm.  Just use a wet paper towel and dip it in the ash
before you light the stove.  Residue comes off like magic!  I usually
then use another wet towel to rinse and a dry towel to dry it before
lighting the stove.  Try it, I guarantee it works!

Washing Soda - Fireplace Soot Remover
http://www.care2.com/channels/solutions/home/346

Stove glass cleaner?
http://ths.gardenweb.com/forums/load/firepl/msg120837486989.html

We use the generic version of the Mr. Clean Magic eraser to clean
ours. It works well if the build up is not too thick.