Is It Possible To Win 2-Player Agricola Without Taking Wood?
Normally I wouldn't like to answer my own question, but I have played a lot of online Agricola since posting this one, with the specific goal of taking as little wood as possible, so I hopefully have some interesting new information to provide by now...

Wood is a really hard commodity to ignore in Agricola, just because it's so versatile.  You need wood for all the below things:


to have space to grow your family quickly
to grow your house to a good size so you can get big points from removation at the end of the game
to build early stables and/or fences to pursue the animal breeding strategy
to build a giant pasture in the late game to avoid losing big points from unused spaces
many, many good Minor Improvements require wood, as does the Well


When it comes down to it, wood just does more than anything else in this game, so much so that I think it's a bit of an insult that the Joinery also exists: wood does all of this AND you can get bonus points from having a surplus of it too?  (Not that anyone ever buys the Joinery, because in practice there's always something else useful you can do with your wood.)

In the light of all this, how can you do well in Agricola without wood?  Well, here's the big things you can do without it:


get a lot of food


None of the efficient food-getting strategies in Agricola requires a single piece of wood.  And while food-gathering is not an end in itself, it does mean that if your opponent is hogging the wood space, you have every opportunity to cut them off from a good food engine, specifically by taking all the clay for the first half of the game.  When your opponent has no good way of feeding his family (Fishing and Day Labourer don't count), then it's academic that he can grow his family - fear of Begging cards may mean he just can't afford to.  Lack of a steady supply of food also limits the number of Occupation cards he can realistically play - after the first they do start inhibiting the ability of a food-poor player to feed his family!

So would I recommend a food-denial strategy as a counter to a player who remorselessly goes after the wood?  Yes and no.  In the first half of the game you can definitely hold your own against the wood-mad player by trying to starve them.  Unfortunately, you have to be an actually terrible player to stumble blindly into a situation where you can't avoid getting Begging cards; so generally I'd say the food-denial strategy can only contain, not crush, an opponent.  It's not hard, though, to engineer a situation where you have a Fireplace, some grain fields and maybe an Oven too and your opponent has nothing but a lot of wood and a heavy reliance on Fishing and Day Labourer.

The problem is that I don't think this position is sustainable all the way to the end of the game.  You can't cut an opponent off from all avenues indefinitely.  I had a great game where opponent dogmatically took all the wood and I dogmatically took all the reed, every time.  Sadly he finished a few points ahead of me: the sad fact is that, even once I buy the Basketmaker's Workshop, a gigantic pile of reed is only worth 5 points and some food, while a gigantic pile of wood enables, at the very least, the building of a giant pasture that uses up all the empty space in your farmyard and ends up being worth 10+ victory points.  Plus, to add insult to injury, they probably go on to buy the stupid Joinery and totally crush you!

As such, my ultimate conclusion is that, no, it's not possible to win Agricola without ever taking wood.  Not without some ridiculous combination of Occupations and Minor Improvements, at any rate.  There are good ways to attack the players who take wood aggressively at the start of the game, but once you've done this you'll want to make a strategic U-turn and compete for it aggressively yourself starting from about round 7 or 8.  

Wood is just too useful for too many things, and if you've played a very miserly, explosive-growth-impeding game, you still need 12 wood at the end to build a 3x3 pasture and save yourself from losing double figures of points at the end.  What's the alternative - ploughing 9 fields?  I don't think so.