You might not live near the ocean like us, but chances are you live near water of some sort. A foot or more of water is essentially all you need to kayak. You need a kayak and paddle as well, of course. We own our kayaks because we use them frequently, but if you want to kayak you can either find a place right on the water that rents them or you can rent them from an outdoor store and travel to where you'd like to go. If you're kayaking with another person then I'd suggest separate kayaks, not a double kayak. Double kayaks are lovingly referred to as "divorce boats". Don't go there, just get your own! However, I have heard of two adults using a double kayak and then sticking a child in the storage hatch in the middle- sitting in it just like another kayak seat, not closed inside, of course.
If your child can sit up and wear a life jacket then they can sit in your lap in calm water while you kayak. We have a beach near us that has a little island offshore, so we like to take our son there because it's a journey with a destination that's not too far and then we can also explore the island. I think it's a nice family activity. Get out there and try it. Have fun and be safe.