Flowering and fruiting (tomatos, squash) are part of the reproductive process for plants. Thus, they require a lot of energy. Where do plants get energy from? The sun via photosynthesis. So the general answer to your question is "no".
Most guides will tell you 8 hours of sun minimum.

I disagree 100% ..there are many edibles you can grow in shade..I live in Florida and am not so familiar with what is available to you in Virginia but mushrooms come to mind! rhubarb? celery? lettuce? blackberry, raspberry, radish, pumpkin, peas, turnips, hops, kiwi..the list of those able to tolerate shade/partial shade and even thrive in shade could go on and on forever..people get hung up on traditional crops..think outside the box :) I remember seeing wild blueberries always growing in the shade in Va..Good Luck!

Here is a link that might be useful: Edibles you can grow in shade

Just occured to me ..if you can grow azalias there and it's nice acidic soil blueberries really would love that..they love acidic leafy fluffy soil..and if there is enough sun for the azalias there should be enough sun for them..down here we do southern highbush or natives ..what types do you do up there..um..rabbiteye? highbush?

Mint for tea might be easy to grow. As well as nasturiums for color (flowers) and peppery flavor (leaves) in your salads.

We have acid soil and humidity although we also get sun. We put hydrated lime and coral shell in the garden around the plants that aren't fond of acid soil. You could try amending the soil so it isn't as acid.

Small fruited tomatoes might work better. We grow them because of fruit flies, might also work for shade. You could also try putting a bag over the fruit after it's been pollinated but before the squash borers get to it.

Any sort of air movement might help with the powdery mildew. Spacing the plantings further apart might help.

Muscadine grapes may be an option since they are native and grow in your local woods. They would like some sun but will tolerate shade. They are almost always growing in oak trees.