Flax Seed is sold by Herm from his second tier, 12 at a time. Four seeds plant a Garden Bed of Flax, with Farming mastery 5. Until Flax Thread gives you the Spinning Wheel, though, flax itself is only an alchemy ingredient.
The seed has a use of its own: 4 Flax Seeds press into 2 Seed oil at the Millstone, which goes into Melted Fat and Fish nuggets.