In a very long association with old puppets, and with much reading, I am of the opinion that wood and papie mache are the only contenders for the pre- Victorian period.
If you wish to supplement the scarce information we have on puppet materials with that from the history of doll making then this tells the same story.
Several oil paintings of the early seventeenth century depict noble children playing with wooden dolls dressed in finery. Wood is the oldest material used for making dolls for sale. Turned dolls were mass-produced from the seventeenth century. Wooden dolls were very popular throughout the eighteenth century. The head and body were carved by professional doll-makers and the face and hair were painted directly onto the wood.
A mixture of paper, sawdust, plaster and glue, called composition, was developed around 1800 as a cheap alternative to wood - later than the period you enquired about. It could be formed under pressure, allowing for the mass production of dolls.
However since puppets (other than toy puppets) were never mass produced like dolls I don't feel that the doll techniques are entirely relevant.
But wood was available, plentiful, and labour was cheap, and as the body of the puppet - the jointed marionette - had to be wood - then one must surely assume that is what the whole puppet was made from. Certainly from early Victorian times cloth and papie mache were combined with wood in an attempt to make heavy puppets lighter. You got upper arms and legs made of calico attached to carved wooden lower parts, for example.It is possible that such techniques were used earlier.
It's good to squawk!