Can multiple users run Farm Report on a single computer?

Yes, do the following:

Put Farm Report.jar in a location visible to all users (for example, C:\Users\Public\Desktop on Windows)
Each user who wants to run it can double-click it from the shared location (for example: their desktop)
Each user's data will be written to the Farm Report.db file in their default folder (on Windows: Documents, on OSX the user's home folder)