Quick update in case anyone else comes across a similar issue:
I was able to install the zorin-windows-app-support package after experimenting with the PPAs. I had noticed that for some reason apt (and Synaptic) was not seeing a majority of the packages from the repo. Not sure what would cause this as I'm not familiar with the way PPAs work.
Remove the ppa:zorinos/stable repo through Software and Updates or with sudo add-apt-repository --remove ppa:zorinos/stable, update apt, re-add the PPA with sudo add-apt-repository ppa:zorinos/stable and update apt again. All packages became visible and I was able to install Zorin Windows App Support through the Software Store.