Could you find a word -- a noun -- for a person who uses few words?

The short answer, yes. There are a number of attested words from which
to choose.

The OED gives the following:

breviarist, one who writes a breviary or abstract.

breviator, one who makes summaries or abstracts. Also ABBREVIATOR.

compendiarist, a maker of compenduims. COMPENDIUM, 1. a short cut, 2.
an abridgement or condensation of a larger work..., b. an epitome, a
summary, a brief, c. ...an abstract.

compendiator, under COMPENDIATE, to sum up concisely, hence, one who
compendiates.

laconic, B. sb. (the adj. used absolutely) a laconic speaker.

laconist, one who imitates or takes part with the Lacedaemonians
(Spartans or Laconians).

Both laconic and laconist derive from Laconia, the ancient Doric
state, the capital of which was Sparta, the citizens of which were
noted for their terse speech. Thus, one who is sparing of words might
be called figuratively a Laconian or a Spartan.

Source:

Oxford English Dictionary, Compact Edition, Oxford University Press,
New York, 1971.

hlabadie-ga

The best descriptors I can think of are the adjectives "laconic" and
"taciturn," each of which lacks a noun form referring to a person who
possesses the quality.

I have seen "Trappist" used in the sense of "man of few words."
("Trappist" more commonly refers to a person who has taken a certain
monastic vow of silence.)

Likewise, I've occasionally encountered "He's a regular Coolidge" and
similar constructions, referring to the tight-lipped former US
President Calvin Coolidge, nicknamed "Silent Cal."

Similarly, you could call someone a "Garbo" -- Greta Garbo was the
woman of few words in her day.

Some great vocabulary suggestions by pinkfreud!  

I'm just a simple country mathematician, but I've heard folks who
don't say much referred to as "bumps on a log".

regards, mathtalk

succinct
terse
concise
trenchant

no noun,  ratz!

How about "introvert"?

Ace,

I am an introvert, but few who know me would say that I am a person
who uses few words. ;-)

~Pink

pink,

If you're an introvert, I'm a nun in a cloister who has taken a vow of silence.
:D

I was unable to find any noun as you describe, although I found
several dozen verbs that have the meaning you are looking for. The
only dictionary definition of a word being a noun that has anything
like what you are looking for was "silent" but in this case it refered
to silent movies, as in "I'm going to the theatre to watch a silent".
I wish you the best of luck with your question.

mute

mmmp

Depending on context:

churchmouse
clam
silent partner

economic

How about "lexiconomist" or "lexinomist" for an answer?