Please, no maybes and probablys and iircs. Mythical easterlands of amazing ninjutsu samurai warriors, however romanticized, does not work as a basis for you to say 'maybe probly it existed'.
Swords don't work that way you propose. Cuttable sheaths? Did you actually just say that or am I going nutters again? I feel sorry for you that this will be offensive to you, but that was by large the stupidest thing I've heard this year. Nothing in that notion made sense.
A) scabbards are not only for transporting your sword, but also for protecting it. To have the sword cut through it completely defeats the purpose.
B) No samurais actually carried their swords in the back, even in situations where they carried multiple swords, they just slung all of them on their side.
C) Swords are not light sabers. If you've handled one, you'd know this. You can't just easily cut through a material with one swing without drawing or hacking the blade in some way. Even if that said disposable vinyl scabbard existed, you'd still have to DRAW the sword from the said scabbard to apply its cutting power to the material designed to be cut. Then again, if you can draw from your back, why use a disposable scabbard?
The ironic part about that is that that is potentially the stupidest post I've seen all year. Being such an asshole while also being so completely wrong is definitely a contender.
A disposable scabbard would be rather easy to make, and you clearly have no idea what you're on about. First you start off with a mostly normal nodachi scabbard. Then remove a slice on the narrow side that faces the blade, and wrap a thin cloth or material of some sort around the scabbard. Swords don't rattle around in scabbards, they're fairly tight, and hold them snugly, so it shouldn't move around enough to cut the fabric until you apply sideways force to the hilt with your hand. Boom, disposable back scabbard, and all you have to do is replace the cloth after you draw it. Not exactly ideal, and you might need to experiment a bit with types of cloth to find one that cuts smoothly but the concept would work. Now whether or not it would be practical enough to actually bother doing is another question, but you could definitely make one.
Also, I'd recommend injecting a bit more courtesy into your posts in the future, because you seriously sounded like a condescending asshole.