How can I convert a shadow box fence to a solid fence?
The feasibility depends on a couple of things:


If you can remove the existing boards without breaking them. My guess is probably, but you would need to do it carefully. I would use a short 2x6 and a light sledge to tap on the board from the backside. 
What the spacing is on the boards. If you can just take the boards off one side and put them on the other, that makes things easier. If the spacing is wrong for that, you would need to pull the boards off both sides and then redo them.
Whether you are willing to do the work yourself. If you are paying somebody, it may be cheaper to have them tear down the old ones and put up new ones than to carefully remove the old ones and reinstall them.