Vegetables that grow in shade are predominately leafy vegetables and root crops. Vegetables that produce fruits, like tomatoes, cucumbers and eggplant, really need all the sun they can get.
If you are going to try growing vegetables in shade, remember that they still need plenty of water. And water and shade are perfect conditions for snails and slugs. You'll need to be especially diligent in scouting for these slimy creatures, or they will wipe out your harvest.
1. Salad Greens
Leafy salad greens are staples in the spring garden. They like the cool, dewy days early in the season and you can succession plant most of them for a long harvest period. It gets a little tricky to keep them growing in the hottest part of the summer. They are slow to germinate in hot, dry weather and quick to blot to seed when they do grow. You can wait until the temperatures cool again, in the fall. Or you can take advantage of a shady spot in the garden, even behind taller corn or tomato plants, and keep them going all summer.
2. Cooking Greens
Leafy vegetables grown for cooking, rather than salads and fresh eating, will actually grow slower and more tender in afternoon shade. They probably won't get as large as their full sun versions, but the smaller "baby" leaves will require less cooking and are often sweeter. (3-4 hours of sun per day)
Most root vegetables can get by on a half day of sunshine, however they will grow more slowly and take longer to reach full size. You can harvest some of them, like carrots and potatoes, while they are still small and sweet. You can also harvest beet and turnip greens, while you're waiting for their bulbs to fill out. You'll want your radishes to grow fast, so that they don't become woody or overly hot, but partial shade will prevent them from bolting to seed. (4-5 hours of sun per day)
Many culinary herbs are fast growers. They will bolt to seed quicker than lettuce, in good growing conditions. They may get a little leggier, when grown in partial shade, but since you're growing them for their leaves, it doesn't really matter. These 6 herbs will do fine with only about 3 hours of sun per day: chives, cilantro, mint, oregano and parsley. (3 hours of sun per day)
5. Peas and Beans
Peas and green beans like cooler temperatures. They need some sun to produce flowers and pods, but they tend to fade out as the temperature warms. Planting them in a cool shady spot will lengthen your growing season.
Bush beans are a better choice for shade than pole beans. Pole varieties start producing beans later in the season and they need sunshine to grow the vines that will eventually hold the beans. Bush beans are quick growers and, like peas, appreciate a little cooling off in the afternoon. (4-5 hours of sun per day)