Yes, you can. I do it - cardboard buckles after a lot of use. I found using thin foam (craft foam, or Foamies) gives a good flexibility to the mouth but it's still sturdy.
You can actually see how I do it here:
http://www.schoolofpuppetry.com.au/tutorials.php/how-to-make-a-muppet-type-puppet-human-foam-puppet-patternYou can download the instructions/free pattern for making the mouth using foam.
I should mention too that there are many other materials you can use, but one of the favourites mentioned by members here is gasket rubber.