Well, this is in an interesting question. First of all, there's no "The Amish"...that would be like saying "do the non-Amish go kayaking?" I mean, you have a range of interests, hobbies, skills, etc among the non-Amish, so the same could be said for the Amish. A better question would be: do some Amish go kayaking? And the answer is "yes!" You definitely may see Amish kayaking on a river or lake near one of their communities.
Reader Katie captured this photo near Rocky Fork Lake in southern Ohio, close to the large Amish community in Highland County. In general, the Amish are avid outdoorspeople, and summer brings ample opportunity for outdoor activity. During the warmer months, you'll see plenty of Amish towing boats, canoes, etc on their buggies to the nearest body of water. Fishing is almost a favorite.
I love this photo that Katie took because it illustrates so well what I am saying about the Amish and outdoor activities. Looks like an Amish person decided to strap a kayak to their buggy roof (why not?) and head to the nearest waterway. So Amish kayaking in this part of Ohio with plentiful rivers and lakes would be ideal.
This is a photo a reader submitted years ago that shows a boat being towed by a buggy. So, yeah, summer is a time for indulging and enjoying the outdoors and being on the water is a favorite way to do that!
This is a popular fishing spot at Indian Lake in Ohio. Lots of buggies gathered at the water's edge.
Interestingly, one of the reasons an Amish community exists in the state of Montana is because there were some Amish men who loved the thought of moving to a place where they could fly fish, salmon fish, and go hunting for big game. So, in short, yes, many Amish love the outdoors and activities like kayaking and fishing are accepted and permitted!