Title pretty much says it all.
I have a lot of mature oaks and hickory in my yard of maybe 1/2 acre.

You'll have a rough go of it; most garden veggies want full (or almost full) sun.  But it doesn't hurt anything to try... seeds are cheap.  Next spring I'd just try a few test plants here and there, no need to till up a large garden spot, just turn over a few shovelfuls of soil, plant something, and mulch around it - treat it like an ornamental plant.  That'd give you and idea of whether it would be worth expanding over the next few years.

Some plants will still produce, just not nearly as much and they won't be as vigorous as plants grown in full sun.  But even a few fresh veggies are better than none, and you might be pleasantly surprised.  Especially since, with only a few plants, you're more likely to coddle them a bit.  A lot of folks make the mistake of starting out with too big a garden.  It gets to be too much to keep up with, doesn't produce well, and they walk away with the impression that gardening is just too hard and not worth it.

I planted a two gardens in the middle of the woods.

Corn was a big flop.  Tomatoes, bell peppers, squash and cukes did well with the moderate sunlight I got.

Beans, were rockstars, but they had the most sun of any of the plants.

I considered planting some tomato plants alongside the house with the belief that they also get the reflected sun light and heat off of the house.  I had a great yield of tomatoes this year on my open hilltop garden.  My upsides were truckloads of fresh 3 year old cow manure compost, full sun, plenty of regular rain.  My downsides were I planted too many, too close, didn't keep up on pinching off shoots per the advice of a coworker.  i did get a greater yield than in years past (on other property) but it was a messy jungle and I had a fair amount of wilt on my branchlets.  Still got plenty of fruit though.

Next year I will do less plants, more spaced, back to tomato cages versus stakes due to lack of depth before hard rock.  I will also try to get them started indoors too, to get an earlier jump.  I hate when most of the yield is in the fall under threat of frosts.  We had a frost warning last night but didn't get it.  I still have probably 15 pounds of tomatoes not yet ripe.

anyways that's my suggestion if you have a suitable location on the sunny side of your house/garage/shed.

Google "Best Vegetables To Grow In The Shade."

Mainly greens, IMO.

My garden is 12x102 with the mid third shaded so I figured it would be just right for leaf lettuce, leafy greens, etc.

Turns out my searches suggest the same things.

Good luck.

I have an almost identical issue. I live on four acres but my lot is only 200 feet wide. I've cut down well over 100 trees to expand my yard and crest a garden. Even with the trees down, the tall oak and ash trees on the side of the clearing  block a majority of the sun. I'm hoping to have a garden by late spring early summer of next year. I'll keep you all posted on its progress.

Try it............. Just plant a variety of stuff and see what grows. We got a little over 500# of tomatos this year and they get less than 7 hours of sun a day.

Try sweet potatoes, beets, fennel, green beans, garlic, and herbs.  Those should do well for you there.

In my many years of gardening, I have learned not to try to fight two opponents:  Shade and established tree roots.



You will work like a dog and always lose the battle.