If your firewall (or other) is blocking the activation of PDF Architect, you'll need to create an exception.
For account-based licenses:
The activation server for PDF Architect is 64.15.159.224.
api-updateservice - 64.15.159.222
download server - 64.15.159.251
DNS name: https://oauth.pdfarchitect.org/
For key-based licenses:
The activation server for PDF Architect is 64.15.159.224.
DNS name: activate.pdfarchitect.org
Allow this IP to pass through Port 80 and 443 and you will be able to activate the application.