I think you got the name right-- I think they're called stitching dies, at least when they are special shapes. (For circles/squares/etc., I'd have just called them pierced.)
Papertrey Ink has some, though I don't think they have a duck per se. Here is a
bird die they have.
Other than that, I'm not sure who's much into stitching. MFT maybe. Lawn Fawn maybe.
Any of those thin dies, though, should be able to cut a decent felt without too much extra effort.