Who made me? What do people call me?  

It could be....
an extremely abused saarinen executive armchair with wood legs...?

It is
Saarinen... probably re-covered at some point.

Pearsall Perhaps?

........
It's Saarinen, see legs etc. Current retail price is 1750 in stock fabric.
This might clean up (professionally) okay, but it can't be un-stretched as far as I know, without re-covering.
It could certainly be worse after forty some years of hard institutional use though!

My opinion
I would keep the legs and swap them onto a metal leg chair in better shape (assuming that can be done).

Metal legs
Yeah the metal chrome legs are better looking than bent wood. It's in bad shape but the cushions seem to be ok.
Next off, I'm about to take a tour of an old high school. From what I hear, there's an area of storage that houses a lot of chairs. hmmmmm FINGERS CROSSED!

it's saarinen but needs work
it is the real thing except it is in poor shape. it will need a lot of work to make it in decent shape and it could cost as much as buying a brand new one (approx. $1500 or possibly more depending or where you are). cost would be labor mainly, replacing and sculpting new foam, sewing and installing upholstery, buying new fabric (by knoll preferably or maharam) and refinshing the legs.
also, the chair's best view is as seen from the back, where the back piece wraps around the seat. the wood leg is much more sculptural and more an integral part of the chair than the metal legs option. you are correct in describing it as closely related to the eames/saarinen organic furniture selected and exhibited by moma in 1940 because this chair is actually closer to the original concept of the winning entry in terms of material and construction (compared to the eames fiberglass/plastic chairs). this chair is also the last of that series (series 70 chairs) starting with the womb chair in 1946 and side chair in 1948 and the arm chair version in 1950. it was delayed because the early material used as form was breaking off in the arms during the prototype stage until the resin,fiberglass came along.
if the original uphiolstery is intact with no rips or holes (the green fabric looks original but looks grimy), you could possibly conserve the chair yourself if you decide to go this route (which is the old vintage look vs. the brand new look) by removing the fabric carefully (staples under the seat and the the hand stiching of the back underside only not the seams at the piping), and then soaking/hand washing in cold water until most dirt gets lifted off. it will most likely clean up but will not look like brand new and some slight shrinking could occur (original fabric might be a wool blend kind). you will need to replace the foam yourself if you want to venture in sculpting with an electric kitchen knife forming it as close to the orignal as possible (you may need several good images of the chair in all angles as reference to get it right) and re-install the fabric back (slightly damp before it completely dries so it will safely pull as you are putting it back together).
this is how i had conserved several of my chairs (dirty but not damaged and restorable, usually left on the curb for trash pick-up) without the high cost, but then again this is all up to you. you could go this route do the work yourself or pay someone to do it for you or just find a vintage one that is in good presentable/useable condition.

Thanks minimoma!
I've made arrangements to pick up the chair next week. Hopefully I can remove most of the grime. It's in bad shape but I enjoy helping the little guy out. He needs to be cleaned!
As for shaping new foam. I've never done that before but this may be a good time to think about it....and possible try.
Thanks for your input...and the motivation.
http://gunnerguidry@gmail.com

more on saarinen chair
forgot to mention, there will be some major vacuuming required too (this is what the dyson is really good for) and if the chair is older, there will be no staples but small upholstery tacks that need to be removed and installed (not the easiest thing to do on a composite material used for the form of the back/arm piece). also, do not let the fabric edges fray too much when washing. use thin batting to even out the foam securing it with spray adhesive. use proper tools and protective gear, have lots of patience, always use common sense and judgement and the most important rule of all, do these only as a last resort and NEVER on any piece that is considered valuable.