Cardboard would not make the body look as smooth and round. I have seen some papercraft pieces that look really smooth. But cardboard has ridges and when you bend it, it remembers that bend and you can't undo mistakes.
As mentioned earlier in this build, Joann's has a polyfoam that would work pretty well. But it is only worth it if you get a 40% or more off coupon. Which come out pretty regularly online and in their ads. As far as I know the professional puppet builders use an open cell foam, also known as dryfast foam. Its possible to order this from a place called "FoamOnline.com" If you do choose to purchase from them, the foam you are looking for is going to be Rectangular, Foam Type: Dry Fast (2.15 lb), Firmness: Medium, Length: 24 inches, Width: 48 inches, Height: 1/2 inch.
Also Ecl has a newer build post on the RPF of what he does now when he makes his Kermits, also look there for informational pictures and guidance from the all knowing ECL. ecl's Kermit the Frog Puppet Replica (using my newest patterns)
Since I'm guessing you're new to puppet building, look at YouTube for some tutorials. Adam Kreutinger is a great channel for puppet building. He explains well to novices getting into puppet building, and his videos are entertaining at times. Adam Kreutinger
I know Adam, I did an attempt of a puppet. Some are too small, some are not really great. At the end of this month, I will make the 13 pointed collar (Sesame Street) Kermit puppet. Same materials, but not really the same. Also you can ask me if you want the custom made 13 pointed template (Png file)