According to this New York Times article, it was a guy named Frank Chambers.  First, researchers used radar imaging of Shakespeare’s grave site at a church in Stratford-upon-Avon.  The imaging indicated that the skull was probably missing.  This led them to an account of a doctor named Chambers robbing the grave in 1794:

“We’ve done lots of research literally trying to pick holes in this story,” Mr. Colls said, adding that the group had looked into the names of Chambers’s gravedigger accomplices, the inns they visited before and after the heist, and the depth to which they were said to have excavated; all the details checked out. “If the grave-robbing account is a made-up story,” he said, “then it’s unbelievably accurate in all its details.”

What I like about this piece is its consideration of whether the researchers’ imaging technique “moved” Shakespeare’s bones, which would mean that the inscription above the grave — “Curst be he that moves my bones” — would apply to them:

Whether the archaeologists beaming radar into Shakespeare’s grave were able to escape the curse printed above the grave depends on how much you believe in quantum physics. Radar waves, like every other form of electromagnetic radiation including visible light, carry energy and momentum, a lesson every schoolkid learns when asking where a comet’s tail comes from: particles of cosmic fluff pushed into a stream by the pressure of sunlight. Indeed, scientists have suggested that spacecraft with giant foil sails propelled by sunlight or powerful lasers might be the cheapest form of interplanetary or even interstellar travel.

One of the ineluctable rules of quantum mechanics (and perhaps journalism) is that you can’t observe something without disturbing it and influencing it in some way. For Shakespeare’s remains to be detected, electrons in the atoms of his bones would have to absorb energy and momentum from the radar waves and then kick it back out. So to see Shakespeare is to give him a quantum tickle. Safely embedded in the ground, the bones might not have moved much or at all, but they knew someone was watching.

Good stuff.

Here is Shakespeare, back when his skull was attached to his body: