I have done this...
...but only on the Plycraft knockoffs. I just saw a photo of the Eames chair exploded (current Dwell mag) and it looks like they're made the same way. There is an inner shell of thin plywood and this is upholstered with foam, welt and leather, then the buttons, or tufting, is added. This inner shell is then wedged into the outer shell.
On the Plycraft chairs, the inner shell is held in place with screws that go through the outer shell into the inner one.
The Eames chair does not have these visible screws and I've always wondered what holds the inner shell in place. Just pressure, maybe? The Plycraft parts stay put pretty well without screws.
It should be un-doable, whatever it is, since almost all upholstered furniture is made so that it can be reupholstered.
hidden clips/ hooks
There are hidden clips and the cushion slides into them. You will need to firmily slide the cushion front to back or back to front(depending which way you are looking at the ottoman). Check out the link and you can see what the clips/hooks look like.
http://www.mancha.demon.co.uk/spares.html
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