I am not sure I understand what you are asking.
If you are asking if the VPN server that you connect to using Tunnelblick records URLs requested by the computer running Tunnelblick, that is possible. You must ask the organization that runs that VPN server. The Tunnelblick project does not run any VPN servers or provide any VPN services; we only provide software.
The Tunnelblick Project does not run any VPN servers. The Tunnelblick web server (at
tunnelblick.net) does record the URLs of all requests that are directed to it (that is, URLs that start with "
https://tunnelblick.net"). However, it does not record the URLs of any requests that are not directed to it. (It can't, just as the server at "
https://www.microsoft.com" can't record the URLs of requests directed to "
https://www.apple.com".)