I'd say liquid latex build up, but it'd look really different to the clay.

Thanks for the suggestions everybody.  It's obvious I may have to start getting into latex.

Duct tape!? Of course you can do whatever you want, but, that does sound a little makeshift. Do you want this puppet to look good?

If you don't want to get into mouldmaking, there are other options.
I once had to make a bat for a tv show, with only a couple of hours' notice.   I found a plastic bag in the dumpster that had been on a costume that came back from the dry cleaners.  I sandwiched the wire for the fingers (coathanger wire so it was springy) between 2 layers of the plastic and spray glued it together.  Then I did a little light spray painting on it with a $2 spray can.  It was still translucent and thin, not like a thick layer of clay. The body had a block of wood with holes drilled in it to stick the wires in, and was covered with a bit of fur fabric I begged from the wardrobe department.  The head shape was carved from some cushion foam with scissors and spray painted, then the bat was put on a stick - if I shook it up and down the wings would bounce and seem to flap.  For stop motion I would use armature wire  instead of coathanger wire of course.
The paint was still sticky when it flapped in front of the camera, and if it had held still it would have looked pretty shoddy, but it was there in time and it didn't require any mouldmaking and casting.
Supermarket plastic bags for vegetables would also work for wing membranes.   Or,  instead of plastic, panty hose fabric might work for the membrane.  I would probably paint some latex onto it to make it more like skin and less like cloth.