I found this on Gwent support, but I have no idea how to set this up. Anyone have experience with this? How do I open ports?:
1) Make sure that you have all ports required for GOG Galaxy opened.
HTTP 80 and HTTPS 443
Also, make sure that these local ports are open (they are required for all Galaxy components to work properly):
9977 and 9978
2) Such situations may be caused by your antivirus blocking connection.
Please add GOG Galaxy and GWENT files to the whitelist of your antivirus or just disable it.
I had a lot of connection issues including this one for months and I tried everything, but the latest GOG update (which I think was last month) solved everything for me. I haven't had a single issue since, maybe try to update GOG if you haven't already and DO NOT exit GOG when Gwent is open.