Must the Waybox be connected to my broadband/WiFi to charge my electric vehicle?

No, you need a Wi-Fi or Cellular (Waybox Pro Cellular) connection only to use the smart charging features. Without a data connection access to the smart charging features is not possible. Without the data connection plug and charge is still possible.

The Waybox will act as a standard charging device in the absence of a data connection, preset charging limits will remain even if the Waybox is not connected. 

Wi-Fi or cellular connection allows you to set charging plans/times so to avoid your electric vehicle consuming power at peak times, taking advantage of the utility time schedules (TOU) found in the app settings.

Maintaining your Waybox data connection ensures that you will continue to receive ‘Over The Air’ updates, meaning your device always has the latest Firmware.