Online it says "mentions atv trails" but exactly where in the shawnee has atv trails?

Go to the rangers office in Vienna. They will give you a map of the trails. They are color coded as to what the trail can be used for.

Its hit and miss. Tons of trails dont allow it, but many have ATV routes. You would have to search individual trails for allowed options. Same goes for horses. Maybe calling would give more direct answers? Someone might be able to give you the names of the ATV accessible areas. Have fun!