It seems to me that most of those sun loving plants are 6-8 hrs of sunlight. You could try it this year and make sure the taller plants (tomatoes) don't shade the shorter ones.
Peggy

I would be surprised it you have luck with veggies. It doesn't hurt to try with a few. You could still try beans and cukes by seed.

Your garden sounds almost identical to mine. We built a raised bed on the side of the garage - ours is to the east. It gets maybe four hours of sunlight a day (plenty of daylight though). I have been successful with tomatoes, green beans, peppers (green, red, jalapeño) herbs, snap peas, lettuce, carrots, cukes, zucchini and plenty of annuals to attract the beneficial insects.

I tried onions and potatoes my first year with no luck...but wasn't a "well-versed" veggie gardener at the time.

Hope this helps!
Kristy :)