I'm growing spinach, and swiss chard, but they don't seem to be sending up new shoots. Can these plants along with other leafy greens push up through wood chips?
New small leaves of tender plants like spinach don't have the strength to lift a wood chip. Neither does the new growth often make enough sideways progress underneath a piece to snake its way up between chips. A good weed like bindweed sure will, but not the wonderful wimpy plants we like to eat. That's why it's not good to plant seeds under chips.
On the other hand, chips even an inch or two away from established plants, while one can argue about their overall benefits or lack thereof, won't cause new spinach leaves to fail to emerge. That's because new leaves in spinach emerge from the center of the plant, like lettuce or carrot leaves but not like rhubarb or peonies. So as long as the wood chips aren't right on top of the spinach but are an inch or two away like you describe, that isn't why they aren't producing new leaves. As to what IS causing them not to grow I can't tell. But it sounds like the wood chips aren't to blame.
There's an argument, to my mind excessively clinical, that wood chips bind nitrogen in the short run, robbing the plants they surround. Eventually as the chips rot from below, the nitrogen gets released again, together with wonderful new humus for a net long-run benefit. I use lots of chips, mostly around ornamentals, and I don't worry about it. But some people do. In that extreme instance, your existing spinach leaves would be yellowing etc. And there are many other possible causes of yellowing leaves even if they are.
Spinach hates heat. Is it getting summer-hot in your area, and what spinach that is growing shooting straight up?