You could actually fuel a Weber Kettle with wood. I would go with small chunks, not long logs or splits. You would have to start your fire early, and get a bed of coals glowing, and add small amounts of wood to keep it going. Or, you could get some wood burning in a fire pit, and add hot coals while you cook.
The key is to be able to have a good bed of coals on the Weber, and add fuel as needed. If you are adding raw wood, you'll want to add small amounts at a time.
So, yeah, I think it is possible, but you will have to plan your cook, and keep a close eye on it.