There is network stuff in .Net VB or C#. I mainly use C# but the same stuff is available in VB, it is just different syntax, both use .Net.
For TCP on a network I just did the code here and it worked well on 2 PCs on a home network.
For a web solution I think you will need access to a network server to use TCP, or you could maybe use a file based system, but all the firewall security stuff might be complicated, I just don't know enough about it, but maybe look into it.
Here is another site using TCP/UDP. Not sure how it could work over the net (PCs not on the same network) without a dedicated server, but that's possibly just me not knowing much about it. I think somewhere, one machine (server) must have an IP address that is web accessible and you have significant access to.