LEGO Digital Designer does not to connect to the Internet on Win10 laptop
Quoting from Lego's LDD page:

Sorry. We don’t support LDD version 4.3.10 anymore.
If you still use version 4.3.10 you will experience errors when launching the application.
You can still use “offline” mode if you wish.
We urge you to update, use the download link, and install free application for Windows PC &amp; Mac OSX.

There is nothing wrong with your computer, or your internet connection (and it is not just you). LDD will still function normally, albeit in &quot;offline mode.&quot;
LDD is now past its end of life, and Lego apparently does not care any longer about collecting usage data on LDD.
The &quot;Internet access failed&quot; message is because LDD can no longer reach a Lego server (62.199.219.133). LDD &quot;phoned home&quot; every time it was started. Now because it can not reach its home server LDD generates this &quot;Internet access failed&quot; message and will only run in offline mode. Apart from spoofing the replies LDD is looking for there is no way to run LDD in online mode any longer.
However, seeing as how Lego shut down the DesignByMe service you are not missing much (if anything) by running LDD in offline mode.