The Official Atari Lynx Club run in the mid-1990s in Australia. The club could be joined by filling out a form that was included in some of the Atari Lynx console boxes. Upon joining members received a selection of club merchandise including a pin, wallet and an Atari sew-on patch.
Thanks to ausretrogamer for the photos of the T-Shirt and Cap.
Other merchandise included a T-Shirt and a Cap. A mug with the club logo has also been seen around but a clear image of it is unfortunately not available.
There was also a tote bag bearing the Lynx Club logo.
This is what the membership form looked like. Thanks to Juan José Torres for the scan.