Following Bev's suggestion..you can try putting some holes about an inch to 2 inches from the bottom of your container, then put a layer of clay rocks, then your usual layer of soil. That way when you water, the water will stay at the bottom, will be a reservoir of water for the plants. The holes will act as a spill over point, so it will not oversaturate your soil above, and still provide air around the root zone. With the plastic container you have, you can visually see also the level of water pouring below.
The clay rocks can wick the water too from the reservoir, so it will keep your veggies roots moist.
I use this set-up, though without soil, just clay rocks on a plastic container with two spill over holes for my spider plant, and so far it works. Maybe experiment with a small container and see if it can work for your veggies before you implement on your entire set-up.