Have a skilled craftsman make one for you out of top-grain leather.
It is foldable, portable, and complements a nice set of wooden pieces.
You want full-grain leather, not top-grain. Top-grain is junk. "Top" in this context doesn't refer to "top quality", it refers to the top side of the leather, which gets sanded away and then an artificial grain pattern is pressed into it. It is a method of making low-quality and/or damaged leather look better.
There are two ways to make a roll-up leather chessboard. The first way is to make it the same way as roll-up vinyl boards are made, i.e., with the checker pattern printed onto the leather with ink, like so:
The second way is to stitch individual squares of different colored leather together, like so:
For the stitched method, any competent leatherworker could probably do it. For the printed method, you need someone who knows how to screen print on leather, and is set up for printing on something 20" x 20" (which means they need a big screen, and a method of making a big film positive, and a big pallet on their press (sign printers will probably have all these things, with the likely exception of specific knowledge about printing on and selecting leather, while T-shirt printers probably won't).
Lately I've been considering making some leather boards (screen printed). I have a background in screen printing, and in leather, though I've never specifically screen printed on leather. Through research I've been able to find out what ink works best on leather, what mesh count screen to use, and various other things, and I don't think I'd have any problems doing it. The only question is: is there enough demand for such a thing to justify printing up a batch of them?
Have a skilled craftsman make one for you out of top-grain leather.
It is foldable, portable, and complements a nice set of wooden pieces.