I have a carpenter's pencil that's getting pretty blunt and I've realised I don't know how to sharpen it.
How can I sharpen it without buying a special dedicated tool for it?
Whatever you have handy. There are many tools that would work, besides a pencil sharpener dedicated to this.
The point is, a carpenter's pencil is mainly wood, and can be shaped by any tools you have in reach that can also shape wood.