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I've modelled a few substituted benzene rings in avogadro and retrieved the partial charges of each atom; I've tried to sum the partial charges of the carbon and hydrogen atoms at the ortho/para and meta sites to see the overall partial charge at that site but results don't make sense as the para site always ends up being 0.
How can I calculate the charge at the sites?
Summary: The method used does not handle π-conjugation
The partial charges in Avogadro (and many other software programs) are assigned by default by the Gasteiger-Marsili scheme†
Importantly, while the Gastieger charges are fairly good for atomic partial charges of organic-ish molecules, they're unlikely to reproduce effects in conjugated systems very well.
A method is presented for the rapid calculation of atomic charges in σ-bonded and nonconjugated π-systems. Atoms are characterized by their orbital electronegativities. In the calculation only the connectivities of the atoms are considered. Thus only the topology of a molecule is of importance.
My suggestion is to use some type of quantum chemical calculation, e.g.:
Even semiempirical quantum chemical methods (e.g. PM7) will allow you to assign charges based on the quantum electrostatic potential around the molecule. They'll be much more accurate at the trends you want.
