You should probably have referred to the system recommendations/requirements information, at for example :- https://www.manager.io/download
“Requires Ubuntu 18.10 (64-bit) / Fedora 33 (64-bit) or newer”
That being said, Tails can run Appimages. Typically, all you should need to do is set any relevant folders for persistent storage, and make sure that the Appimage’s permissions are set to be executable.
You do not need a network or internet connection to run Manager, but it will require to use the loopback (typically “lo” on a Linux system), which is not anything to do with your system’s security.