I have done it before, and there are tons of people at Tidewater Kayak Anglers Association > Home
who do it all the time.
The only real thing you need to be careful of is the current. Do your homework in advance and make sure you know how strong the tide is going to be, and in which direction. The entire chesapeake bay is emptying out through there, and I've been there in a boat many times where it would have been inpossible to do much more than hold position while paddling against the current.
That said, I've been as far as the 3rd island of the CBBT, launching form either shore. That's a very long day and not something most people will do, but when the current isn't bad and the waves aren't too big there's a lot of water to be covered. If you're concerned about how big the water is at CBBT, there is a fairly consistent nightime striper bite at the HRBT (sometimes called the "coleman bridge" on this site and others)... which is far less of a paddle.
I've also never fished in the summer... only the fall/winter... so can't really advise on the best places and techniques. But you have opportunties for drum, cobia, flounder, small stripers and blue, spadefish, sheepshead, and probably some more... all around the bridge. There's also great puppy drum and speck fishing inside lynhaven inlet if you need more protected waters.