Wasps do indeed construct their hives by chewing up wood into a sticky papery substance.
Bees hives are constructed from wax, which is made by metabolism of honey in wax glands from which the wax is secreted in flakes and then taken by other workers to be mixed with saliva to make the wax mouldable.
Once the cells are constructed the wax cannot be broken down and reused. Instead the cells themselves are used over again once the larvae inside has completed development or the honey store has been used up.