I received a funny range of answers when I contacted the various shops, spanning from "of course we do!" to "of course we don't!". I understand both of these responses, as a renter unfamiliar with cold weather paddling can be quite a safety hazard, but kayaking is awesome so we should still do it in the winter.
I've mapped all the rental shops that I could find below, with symbols representing whether or not they provide rentals in the winter! Click on an icon to see the name, address, and website of the shop.
Green = Open for rentals in winter
Yellow = Limited winter rentals (group size minimum or limited days)
Red = No winter rentals (most closed November - March)
White = Unknown (contacted, no response yet)
Disclaimer: This summary is based on email conversations that I had with various kayak rental companies in early December, 2015. Please contact a company or check their website to confirm the details if you are planning to rent. Also, if you're planning to paddle in winter, make sure you're dressed for immersion and always wearing a PFD. Paddle safe!
Update (May 2016): I've now added all the kayak rental shops I can find in Belgium. There are still quite a few to be added in the Netherlands. If you know of a good kayak rental shap that is not shown here, please send me their website!