We study strong field double ionization of a series of organic molecules by making use of coincidence detection of fragment ions. We measure the double ionization yield as a function of pulse duration, intensity, polarization, and molecular conjugation. For conjugated molecules we find strong enhancement in the double ionization rate over what one would expect on the basis of tunneling or multiphoton ionization rates. Calculations reveal a correlation between the electronic structure of the different molecules and the observed double ionization yields, highlighting the removal of electrons from inner orbitals.