Answer
Some users (such as many South Australian schools) require additional network configuration from their IT team to allow Lightning Payroll to properly function. The issue is mainly to do with network restrictions and blocked web addresses that Lightning Payroll needs to communicate with.
Below is a list of all web domains and urls that the program needs to access to function properly.
General Lightning Payroll URLs
https://backup.lightningpayroll.com.au/
https://appbackend.lightningpayroll.com.au
https://download.lightningpayroll.com.au/
https://update.lightningpayroll.com.au/
https://lightningpayroll.ozedidirect.com.au
https://emailsettings.firetrust.com
https://api-superstream.ozedi.com.au/api
https://api-ebusiness.ozedi.com.au/v2
ATO/STP
https://ato.sbr.gov.au/services
https://prod2.ato.sbr.gov.au/services
https://authentication.business.gov.au/R3.0/vanguard/S007v1.2/service.svc
https://softwareauthorisations.ato.gov.au/R3.0/S007v1.2/service.svc