I store my data (including my vault) on a mounted partition rather than my home folder. The Obsidian Flatpak is not able to access, or even see, these partitions by default. Which meant that I couldn’t access my vault even though my users had permissions to access the files and folders.
I found the solution and thought I’d share here in case anyone else was running into the same problem.
The official solution is to run the following command to give full filesystem access to the Flatpak application.
sudo flatpak override md.obsidian.Obsidian --filesystem=host
This unfortunately didn’t work for me.
Continued searching lead me to Flatseal a GUI application for managing Flatpak permissions. Through this I was able to successfully apply the permission change that I couldn’t apply via the cli command.