A two treadmills side by side power adjacent paddle wheels on either side of the little craft. The pooch's front paws rest on an angled carpeted slip resistant shelf that gives the dog something to push against while it's walking the treadmills and powering the paddle wheels.
If it wants to turn
left, it walks moves his walk to the right treadmill, but could have the drive shafts reversed so a right turn is facilitated by running on the right treadmill.
Or could simply have one treadmill / paddle wheel arrangement with the front paw assembly being steerable like a bike's handlebars, whatever would work best. Could steer a rudder or the paddlewheel itself.
Possible names might be "Good Buoy", "Snoopy Sloop", "Muttineer", "Wave Wagger" etc.