Yep baby carrot is fine for food. Any fruit or veggies is fine. Any fruit and veggie waste too. You know, like the stuff you cut and throw away. That's all food for them. But remember, go small on the food in the beginning. Another thing, the worms will need for the food to be almost rotting before they can even eat it. Soft foods like kiwi, tomato, strawberries etc is very easy for them to eat cause its soft and squishy and pretty much ready for them to start eating.
They don't have any teeth to bite into hard stuff. But its still food they just need it almost or be rotting already for them to start to be able to eat it. Experiment, and again, also go light on the food and see how much they eat and how fast they eat it. Any food that remains leave it and don't add anymore until its gone.