hi Zach,

That is a great question. My first answer is that we really don't know, and my second answer is that it depends on what you call a sunspot.
Through the use of a technique called helioseismology, we know that the strong magnetic fields which we can observe at the solar photosphere extended downward roughly 10 to 20 megameters (a mega-meter is one million meters). In general, it appears that they can be a somewhat shallow phenomena, not really extending down as far as they are wide. But studies using computer models suggest that the precursors to sunspots begin very deep below the surface, at the deepest parts of the convection layer, when bundles of strong magnetic field start to rise up from the interior toward the surface (a bit like bubbles streaming upward from a glass of soda). And what we see as sunspots are the intersection of these bubbling up concentrations of magnetic field with the solar surface.

Helioseismology uses some of the same physics as the seismology we use on the Earth. Terrestrial seismology is used to figure out what kind of rocks are underneath the Earth's surface through the study of the sound waves that propagate after earthquakes or explosions. On the Sun, the convection or "boilling" of the plasma in the upper layers generates sound waves, and studying how those soundwaves interact with things like sunspots helps us understand what is happening underneath the sunspots. It is a very difficult topic, but one with potentially a lot of reward. For example, just in the last decade it was discovered by some of my colleagues how to use Helio seismology to get an idea of what is on the side of the Sun that we cannot see from the Earth's, "far side imaging". For an example, see http://gong.nso.edu/data/farside/ -- there are some nice additional explanations on that page as well.