Yes, it's allowed but most of the waters in the nature preserve and the old canal system are overrun with algae, water chestnut, and downed trees making it not great for paddling. There's a kayak launch at the parking lot by the vischers ferry fire department that leads into the Mohawk River. There's another river launch on Clamsteam road right by The Klamr. I've heard of people launching from Clute's Drydock but never tried it myself. The river water has been very low recently.
I wouldn't kayak on the water as Justin stated, the water level is low and covered in algae.
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