If the area is getting 6 hours or more of continuous direct sunlight (not blocked in any way by trees, buildings, or anything), you will be able to grow just about any vegetable to maturity. More than that will cause even better growth, though; 6 hours is about the minimum for healthy growth in a lot of vegetables.
If you are going to have less sunlight than that, here are some suggested crops, as posted in this thread:
Generally speaking, leafy vegetables are the most shade tolerant eg:
And some root vegetables:
- Cole crops do ok with limited sunlight: cabbage, broccoli, cauliflower (limiting sun can even help protect the heads), kale, etc.
- Also related are mustards, which can give your salads extra flavor.
- The chicory family is another leafy crop that would give you variety: radicchio, endive, etc.
Read that thread for more information.
As for the trees taking the moisture, this shouldn't be a problem, as they're in raised beds and will need irrigation in any case. If there are tree roots below the beds, then nutrient grabbing will be more of an issue than the moisture loss.
Be sure you fertilize well for a good crop. Ideally, you can find some good compost and fertilize with that. It builds the soil up as it feeds, and everything ends up much healthier that way.