Does computational chemistry include neutrons in ground state calculations?
Depends on what you mean by 'computational chemistry'.
In the most common sense of the term, where the aim of a given calculation is determination of the electronic structure of a particular system, each entire nucleus is treated as a classical point charge/mass.  Thus, neutrons are not explicitly accounted for, aside from their effect on the various nuclear masses in, say, vibrational frequency analysis or in estimates of the kinetic isotope effect.
If you're referring to quantum models of the nucleus, though, then this statement of yours is incorrect:

... most particles in computational chemistry [are] treated as point charges.

In these nuclear computations, if the nucleons were being treated classically it would be correct to say that they are treated as point particles since, as you correctly note, neutrons carry no charge, and proper representation of both the neutrons and protons is necessary in order to obtain an accurate model.
However, as noted quite correctly by Mithoron in a comment, for quantum modeling of the nucleus, the protons and neutrons are not treated as particles but instead as contributors to the nuclear wavefunction.