Wow... floor tiles! Guess it could work but in my experience, they tend to crack really easy. Not talking in puppet use but when I've used them for floors.

A good home machine and number 16 needle could very well get through them ok I would think. You would want a universal or chisel point. I've sewn through similar thicknesses of other odd materials before with that set up. The only thing is I would not go with the self adhesive tiles if you want to try this. Just like that sticky back Velcro, the adhesive is going to gum up your needle in about an inch of sewing and you'll be sorry!