Does the target device need an internet connection for the software to work?

Yes, the target device must have an internet capability enabled for iSpyoo to record the activities of your child or employee.

The software may function over a wi-fi hotspot, given that the hotspot has no filters or firewalls in place that could interfere with the software.

But, no need to have an internet connection always.  Just some time to let the software flush data to the server where you can read it (at https://my.iSpyoo.com)