I've only ever been sea kayaking in calm sea water, and I have a 15 month old, but I would never even consider taking him kayaking even in the calmest of water. My son has had over two months of swimming lessons and has been in a pool almost every day this summer, but he can't swim yet. And I think to kayak safely anywhere in any condition the athlete must be able to swim at the very least, and should be a strong swimmer, imo.
Can your 15 month old swim on his own? What would happen if your kayak over turned and he went under? What if you were not able to get to him right away? Is your child able to pull/climb himself out of the water on is own? What if your little one decided he didn't want to be strapped in any more and started fidgeting? Kayaks tip over soo easily. Do you feel confident that your 15 month old can take care of himself if the kayak happened to flip over? What if he knocked his head on a rock? These are just a few of the questions that come to my mind. I think even with a life vest the chances of something happening are too great.
I think taking a 15 month od kayaking (even sea kayaking) would be fool hardy at best.