An internet connection is only required during activation, because the eLicenser Control Center needs to validate your activation code and download your license.
You’ll need to reactivate your license to switch between machines: https://www.steinberg.net/en/support/software_reactivation.html
Only the computer that you plan to use needs to be online (for activation only) for this to work. The process is pretty simple, you’re basically just activating Halion 6 again which takes less than a minute.
Alternatively you can buy an USB-eLicenser dongle, which will allow you to use your Halion 6 license on any computer that has it installed without having to reactivate. Just plug it in and it works.