Is 'unwielding' a word, or not?

Not. You might be thinking of one of two possibilities:

1) UNWIELDY, meaning difficult to carry or manage because of size and weight. 
http://dictionary.reference.com/search?r=2&q=unwieldy

2) UNYIELDING, meaning unbending or inflexible; not giving way to
pressure or persuasion.
http://dictionary.reference.com/search?q=unyielding

Hey, Byrd.

I know folks who are both unyielding AND unwieldy.  Maybe they're unwielding!

Ha!

That does it!

I like the word.

Let's introduce it.

Here are the Definitions (Dictionary Editors please note):

Adjective. 

Describes someone who has been (1) disarmed after wielding a gun; (2)
made to look incredibly foolish after trying to assert authority.

Hey Paf! I know a few people like that too. Ya might be on to
something there. So how does one officially coin a word, Miko? A
question for Google Answers, I think!   ;-)

How do you coin a word?
I have seen dictionary sites  - especially bilingual ones -  that ask
for reader support.  Some may be more choosy than others, of course. 
One might have to survey the field and then start with the less choosy
ones, building up a track record.
Second idea: get a columnist and/or talkshow to use the word, maybe Larry King.
I bet he has a long list of people that are unwielding.

As funny and enlightening as this Q&A has been, I would hate to be
responsible for the legitimization of a misused word!

I think byrd-ga should get full credit for pointing out the uncanny
resemblance between the words UNWIELDY and UNYIELDING. They are
undeniable similar in structure and definition, but each are still
definable different.

Personally, I could only think of UNYIELDING at the time, which led me
to believe UNWIELDING surely had potential as a word. But it was
UNWIELDY which had slipped my mind, tricking me into believing in a
phantom word! Oh, the drama!

I honestly don't believe that adding the word UNWIELDING to the
English language would result in any disambiguation, and so for the
sake of clarity, please don't appeal to the Oxford English Dictionary
http://www.oed.com/ on my behalf! : )

However, the whole subject might make for a great poem...