Our composting worms are mixture of species (mainly reds and dendras) selected for their composting ability.
You can do - they will definitely help work through the material.
Cardboard is great for your worm composter (as is a sprinkling of lime mix) A ratio of 25-30% volume of shredded cardboard or equivalent will ensure the compost doesn't get too wet and there is air circulating. Avoid the really shiny boxes like soap powder and go for newsprint, egg boxes etc.
Your wormery composter is particularly suitable for peelings and leftovers of fruit and vegetables, such as potato peels, apple cores, green tops of vegetables, etc.
Do not put intensely spicy foods into your worm composter, such as ginger or an excessive amount of citruses, as well as milk products, meat products, bones, oils, lard and other fats.