Last evening I shared the mobile Internet connection with the TV instead of the main connection and I found the update. Maybe it was a coincidence. LG Store works now.
Software version 03.23.17 and webOS 1.2.0-2717.
Well, you do have a slightly different model.
I did contact LG customer support but they are useless. They don't even know which is the latest software version for that model or if that LG Store should actually be working.
What is the TV model you have? Is it a **LB670*? What is your ISP? Maybe my ISP (RDS) is blocking the access to the update server although the preinstalled apps were able to receive updates. Those could be on a different server.