I know this is a slightly older thread but I'm building another blinking puppet and trying to refine my previous attempt, so I'm putting this idea out there as a tip.
To make the curved material over the eye, I got some stretchy material [it just needs to stretch in one direction at least] and stretch it over a sphere. I'm leaving a gap between the eyelid and eye so it doesn't rub, so you'll need a ball bigger than the eye, for that. With your material stretched over your ball, tape it up at the loose end so your material is nice and tightly curved over the rounded surface [it'll look like a microphone] and use something to harden it up. I used art fixative but hairspray should work too. Do some tests, but anything that sets clear should work. It might get stuck and be hard to get it off your ball, but if it gets a bit creased or wrinkled in the process, just put it back on loosely and give it another coat. Your initial setting should keep it's shape enough so as to not need it be tightly stretched over. Art fixative loosens it up again when it moistens so that knocks out any creases and wrinkles from the removal, but it keeps it's general shape still.
I also did some tests with PVA/wood glue and some rather liberal coats left it solid like plastic.
Then it's just a matter of trimming it and securing it to your mechanism- looped wire or whatever.
Important thing is to use something that dries clear.
I'm gonna use this method for non-moving eyelids in future too.
Hope that helps.