On some pasts questions on this site, I discovered that walnuts release Juglone, which is nasty for the growth of many plants.
I had/have a part of the vegetable garden partly under a Juglans regia, for more shadow tolerant vegetables. Note: still few meters away of trunk, and sun reach for good part of the day the vegetable garden. So I wondering if I can still plants some vegetables, and which one. On past, I had usually peas/green beans, potatoes and celery/parsley under the plant.
Potatoes aren't great near a walnut tree, but the others you mention should be fine, if they get sufficient sun. Vegetable plants sensitive to juglone include the nightshade family, so potatoes, tomatoes, aubergines and peppers are better away from the roots. You can grow root vegetables such as carrots, also squash/melon, if there's sufficient sun in the area, see here http://homeguides.sfgate.com/vegetables-can-grow-near-walnut-trees-54124.html
I have a lot of experience through trial and error with this. My garden is surrounded by black walnut trees. Over the years I've managed to locate a few small sweet spots where the walnut roots don't seem to grow, but squirrels will bury the walnuts without your knowledge, and suddenly, a baby walnut tree will pop out. What happens is that your suseptible vegetables will grow (except peas) normally as if nothings wrong. They'll get to a point in the season when it's to late to grow them again. Then suddenly, the plant will begin to wilt like it's not getting any water, because it's not. Juglone prevents the plant from consuming any water or nutrients. As soon as you notice the wilting, the plant will be totally dead in a day or two. There is no saving it. You can try to transplant it somewhere else, but at that stage, it's very traumatic for an already badly damaged plant. Doesn't usually work.
What can you grow? Some things will grow seemingly without any issue whatsoever and other things will grow, but not as well.
Beans. Bush and pole will grow well. I've read that pole beans don't do well, but I can confirm that they will do fine. Just keep them away from garlic and onions.
Carrots grow without any issue
Garlic/onions grow without issue
Potatoes. I have gotten potatoes every year but they're usually small. They don't do great, but they'll grow. I grow russet and red. Suggest potting them instead if you can.
Squash, melons, pumpkins, gourds all grow without any issue
Corn grows without any issue
Tobacco seems to grow without issue. Year two planting it and so far so good. I know we are talking vegetables here, but it's a really pretty plant when it flowers.
Wheat/barley and other grains grow without issue. They actually thrive under the shade of the walnut tree.
Catnip. Again, not a vegetable but I can vouch for it.
Black berries have no issue
Basil, parsley, chives no problem
Okra. First year trying. So far, so good but it's only about 6 inches tall, so too early to tell.
Lettuce will grow if it has some distance (30 feet or so). Shallow roots help here.
Beats grow without issue
Radishes grow without issue
Celery will grow but slowly. This vegetable takes forever to grow in normal conditions, so start it very early.
Cauliflower and broccoli. I have had no issues with either but have always planted as far away as possible. I know the ground they're in is poisoned by Juglone after losing tomatoes and peppers in previous years, but I still got a beautiful harvest last year. This year looks the same.
Cabbage. Give them some distance and it grows. Again, shallow roots. No guarantee on this one. I had success last year and so far this year they're looking good.
That's all I've got for now but will try to update about the okra.
I say if you're surrounded like I am and don't want to risk the iffy ones i listed, grow a ton of various dried beans, carrots, corn, and herbs. All things that can be stored for the year and almost all enjoy. Grains are great too, but it's a lot of work from garden to bread. I happen to like it. Large pumkin patches are always a hit in the neighborhood. I've done my entire yard before but it's just a novelty. It'll make you popular.
Good luck!
