We calculate exactly the vacuum polarization charge density in the field of a subcritical Coulomb impurity, Z|e|/r, in graphene. Our analysis is based on the exact electron Green's function, obtained by using the operator method, and leads to results that are exact in the parameter Zalpha, where alpha is the "fine-structure constant" of graphene. Taking into account also electron-electron interactions in the Hartree approximation, we solve the problem self-consistently in the subcritical regime, where the impurity has an effective charge Z(eff), determined by the localized induced charge. We find that an impurity with bare charge Z=1 remains subcritical, Z(eff)alpha<1/2, for any alpha, while impurities with Z=2, 3 and higher can become supercritical at certain values of alpha.