Can a polarized object have a net charge of zero?
Yes, this is a common case. A simple example is the gaseous molecule HCl, which is polarized so that the H end is slightly positive and the Cl end is slightly negative, but with an overall charge of zero.
Polarization simply means that the center of charge is displaced from the center of mass. Polarization can occur in positively charged cations, negatively charged anions or neutral molecules. However, the effect of polarization is most important in neutral molecules because the interaction energy of charged molecules with other charged molecules or dipoles is much larger than the interaction energies of polarized neutral molecules.