Resolution
The fix is to go to a PC with a working copy of Legistar and copy the files Legistar.exe, Legistarlaunch.XML, and Icon.XML from C:\Program Files\Daystar\Legistar5, and paste them into the folder on the PC showing the error message.
These files may be in a different location on the other PC. For example, they may sometimes be found in C:\Program Files\Granicus\Legistar5.