In the following post I discuss topics pertaining to the Northern and Southern Water Tribes as they exist after "The Southern Lights" episode of Legend of Korra, so in order to avoid confusion I use the terms North or South Pole to refer to the specific spiritual point rather than the entire landmass. To address the landmass, I state "the surrounding area" or something of the like.
Where's the Wood?
So, we are now introduced to wooded regions in the surrounding areas of both the Northern and Southern Water Tribes. Previous to "The Southern Lights" in Book 2: Spirits, the only vegitation that we see in the surrounding territory of either tribe are the plants by the Spirit Oasis at the Northern Water Tribe, but these are only shoots of bamboo rather than trees.
Both of the forests that are revealed appear with great spiritual importance. The forest at the Southern Water Tribe circumscribes the South Pole and is filled with spirits, and although it is not said whether or not the forest in the North is in the vicinity of the actual North Pole, we know that it is the home of many spirits. Regardless, both of these forests do present a possible source of lumber for each tribe; although, it is quite unlikely that the people of either tribe would take resources from a place with such spiritual connections. Whether it be out of a desire to pleas the spirits who dwell there, or out of a fear of the consequences if they were to disturb the area.
Since there is such a small chance that either tribe would, in fact, cut down trees for wood in such a location, there still exists no logical explaination to where the wood comes from that is used in so many ways by both water tribes. We see wood used in everything from ships and tools, to weapons and shelter, in addition to its basic use as firewood. So, from where does the wood come?
Trade is a quite plausible explaination; however, prior to the reconstruction of the Southern Water Tribe, the tribe was quite secluded and its inhabitants did not interact with the other nations, outside of fighting off Fire Nation raids and sending warriors to help the Earth Kingdom, so the question still stands, where's the wood?