I am wondering if it is possible to carve the butt stock on a red rider BB gun
Sure, just mount it securely to the waste board and carve away. I’d suggest practicing your design on a scrap piece first.
I do guitar necks and bodies. Gun stocks would be similar to instrument parts. For the best results, you probably would want to model a stock in 3D and do double sided machining . This is possible on the X carve with the proper software.
If you are talking about carving something on an existing BB gun, then it is probably just a case of engraving, which you probably have done already.