Simple answer is yes it could be done, if it can be done cost effectively is another question altogether & one only you can answer.
Your 2 main problems are mounting & holding the material on the cnc, making one simple jig is unlikely to work as I am guessing each piece is going to be a different size & shape. You might be looking at having to make something for every piece of material you carved.
I am not familair with the link highpockets added but have seen it mentioned before. A touch probe would map the surface of the material & then allow you to machine what you wanted on to it.
Worth experimenting but the time involved with each piece of material may make it a bit expensive