I'd say yes, but not as strongly, I wouldn't think. As the worms incorporate the wood chip material down into the soils, I think it has a harder time being broken down and steals more nitrogen in the process. Hay and straw are woody but not as woody as wood, so are more easily broken down by the various organisms and microorganisms to be found in the soils.
The faster the breakdown of the material, the quicker it binds with nitrogen and is then neutralized, whereas the long, slow process of breaking down wood chips would likely bind up that nitrogen for much longer and also require more nitrogen for the binding.
Straw is woodier than hay even, so it takes a longer time in a compost pile in which to break down...this I know.
I've used hay around my veggies for years before I ever did the BTE and I never noticed any kind of nitrogen depletion from the use of it. Just good moisture control , weed suppression, and good addition to the soils when it broke down.