Answer: It's a good question because walnut tree leaves and bark do contain compounds that can impede the growth of some other plants.
My sense of it is this: If you new vege garden is not inundated with leaves and nuts, it's probably fine. Right around the base of the tree is clearly another matter, but if you well beyond the outermost branches I do not forsee a problem.
Beyond the growth inhibiting compounds in walnuts, shade and root competition from any kind of tree is also a factor. If your new garden area is not overwhelmed by walnut roots, and if the trees branches do not shade the new garden area, you'll be okay.
My advice is to go ahead and develop your new garden area. With the above factors in mind, if growth does not meet your expectations, you'll have some culprits to consider.
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