I'm not sure how much this will help you, but one person who seems pretty active in creating software to compute homology and cohomology is Robert Ghrist. You can find his website here and an MO post where I discuss his work here.

I'm also including some links below which I've used in the past to do some cohomology computations. These computations used spectral sequences, so this might not help you as much as Ghrist's work, but I figured it could not hurt to include these links. I'm sorry if this turns out not to help:

Bob Bruner's code and tables

Christian Nassau's website for more modern approaches to the same types of computations

I'm not sure how much this will help you, but one person who seems pretty active in creating software to compute homology and cohomology is Robert Ghrist. You can find his website here and an MO post where I discuss his work here.

I'm also including some links below which I've used in the past to do some cohomology computations. These computations used spectral sequences, so this might not help you as much as Ghrist's work, but I figured it could not hurt to include these links. I'm sorry if this turns out not to help:

Bob Bruner's code and tables

Christian Nassau's website for more modern approaches to the same types of computations