I am sure it could be done but why would you want to. If you get a decent fish on you would have to kite back to shore making the caught fish a live bait for something bigger with big teeth. No thanks
Why not try dangling a net from the kite as well and see if you can snare a mutton bird while you are at it ?
Why not tex-screw a flower pot to the front of your board and trying growing tomatoes while you sail along ?
Why not carry a rifle and see if you can down a gazelle whilst boosting big air ?
If you went windsurfing your board would be big enough to light a fire on the front and you could have yourself a roasted garfish, mutton bird and gazelle sandwich with a tomato side salad.
(roasting a gazelle on the the front of your kiteboard - that's not living)