Here's a partial list of materials that I use. But you may come up with other ideas.
Super 77 by 3M - excellent spray adhesive
Hi-Strength 90 by 3M - much thicker spray which I don't like as much as the 77
staple pliers - available from hardware stores, an indispensable tool
gaffer tape - not duct tape which turns to a gummy mess
foam core - I ask local frame shops to keep any scrap for me. I usually get enough pieces to make all the puppets.
open weave fabric- black - for inside the mouth
polyfoam - 1/2" 1/4" 1" thicknesses - I just bought large sheets from The Foam Factory
closed cell foam -
"landau" foam - used for automotive upholstery
plastic Christmas balls
clear craft Christmas ornament balls
large newsprint for making patterns
sharp, good scissors
The Foam Book: an easy guide to building polyfoam puppets
by Drew Allison & Donald Devet at Amazon