It depends on which one. The Second Doctor was the first the use it and so have the 3rd, 4th, 7th, 8th, 9th, 10th and now the 11th doctor. It is never revealed if the Doctor built the earlier models but we saw at the end of "The Eleventh Hour" that the newest one was created by the TARDIS itself. It may be that the TARDIS makes the Screwdriver and the Doctor then adapts and tampers with it.
Also I believe it is the 9th Doctor that indicated that he destroyed a factory that made sonic devices