No you do not need an internet connection to use PyXLL.
The only thing that won't work without an internet connection is retrieving the new license key when your subscription renews, but you will be emailed the key which you can update manually in that case.
After the current license key expires PyXLL will try and fetch a new one (which it will save, so after it has the new license it won't need an internet connection again until next time). If it can't, it will continue to work for a number of days after expiry to allow you time to update it manually (7 days after expiry for the monthly license, or 14 days for the annual license). If after that time there is still no internet connection and the license key hasn't been updated manually you will see a pop-up dialog telling you that the license has expired.