Seating: Paper Cord Replacement  

It has probably been done before. But for details you better ask @spanky .

Are you talking about weaving strips of leather?  I saw a set of walnut Møller chairs once that had seats woven of narrow strips of leather.  I think it was in the usual Møller weave pattern but it was years ago and I could be wrong about that.  But if you mean narrow strips, then yes, it can be done.  It would be the same as weaving rattan, only maybe more difficult as leather isn't as stiff as rattan and may be hard to get through the final rows.  Just guessing.

If you mean doing a seat out of a solid piece of leather, sure.  Upholster it the same way fabric is done on Møller chairs.  The only vegan leather I have any experience with is the stuff Birkenstock uses for sandals, which is too thick for most upholstery.  But i'm sure there are lighter weights available. 

There's a thread here somewhere about how to do fully upholstered seats.  I would do that way before I'd do a sling-style seat of leather (because it's totally incorrect style-wise). 

For authenticity and the correct look, full grain/top grain vegetable tanned leather is what you are looking for.  It is very rarely used for upholstery these days, because it has some downsides.  In the 1950s you basically had two options for leather with danish chairs, especially the high end ones, natural or black painted.  And natural is what becomes 'cognac' color with age and or oiling, but starts out a very unappealing fleshy pink.  You can still get the natural option and if you look long enough you can find it thin enough for upholstery.Black is specialized form of leather paint, probably a nitrocellulose lacquer.  It is almost impossible to get a similar leather now, because what is sold is aniline dyed leather.   In the 1960s the Danes broadened the gamut of available painted/pigmented leather colors, which is also easy to spot on a chair when you know what you are looking at.  Unfortunately most people nowadays just go get some high end aniline dyed, chrome tanned leather or low end pigmented leather in whatever color they want and think that looks fantastic, but it is extremely and obviously wrong if you know what you are looking at and what it should look like.  

Thank you all for your post. After reading your responses, I'm considering having them upholstered. Any suggestions on authentically styled new material?

By the way, this forum is wonderful.