The only question is if I were to use 1/2 Foam how could I make it so it would be not too flexible but a bit sturdy in between?You can also use 1/2 foam (either polyfoam or reticulated).
So basically when I use gasket rubber I just need to make sure I don't leave a space between the middle?I wouldn't use foam. I've experimented with foam mouth plates before, and while they do give the mouth flexiblity, they don't retain their shape very well.
Like Slackbot said, gasket rubber is probably your best bet. Just remember not leave at least half an inch, if not more, of space for the hinge, and when you're fabricating the puppet not to pull the fabric too tight, otherwise you'll end up with a puppet with a permanent smirk/sneer.
They can actually work quite well, but they have to be designed properly and appropriate for the puppet. When using 1/2" foam you often need to create a firmer mouth plate(s) to glue on to the 1/2" foam (smaller than the actual mouth) to help the mouth to retain its shape and to provide something for you to mount a grip for your fingers/thumb on to.I wouldn't use foam. I've experimented with foam mouth plates before, and while they do give the mouth flexiblity, they don't retain their shape very well.