Plastic snaps - I get mine from a large box in the cupboard here ... :-)
Actually, this is a very good question. I bought mine from a large distributor here in Australia - Shann Accessories. But I had to buy 100 sets = or maybe it was 200. They weren't expensive when bought like that, in bulk, and I do use a lot of them.
I had a prowl through the usual suspects like OWFINC and Seattle, but no joy there. I eventually found the source here:
They are called 'suntenac', and I think the ones I use are the 10 mm ones.
Now, where do you get them in Japan or USA? I have no idea I am afraid. You need to check the commercial distributors, whoever they may be.
The alternative is to buy *stainless* steel sew-on snaps. These are readily available in small sewing shops. I even found some in the general store in a small town in France, and bought some on a card. Far less convenient of course, but if all you need is a few it may be the best/only solution. Sewing these ones on means you don't need to make up a punch and die either.
So - can I expect to see a photo in due course?