I imagine similarly could be done on any old computer? (Amiga, etc?)
Yes. I do this occasionally with a TRS-80 Model 100.
And back when they were both still current, I would dial in to a PR1ME and Vax minicomputers with a Commodore 64.
My holy grail is figuring out how to run a personal Zork server on an Ubuntu machine that I can access (serial or telnet) from the 100.
What on earth is a "Zork Server" anyway?
And its 70Hz mono monitor is still decent by modern standards. (If you've only seen 70Hz colour CRTs: the ST's mono monitor is not the same! Mono phosphors look quite different. No flicker, like an LCD, but with restful, paper-like contrast levels, like a Kindle.)
A regular A500 can do TCP/IP just fine without a vampire.
By using AmiTCP or roadshow TCP/IP stack and PPP, SLIP, PLIP (e.g. with a plipbox) or a NIC based on the left expansion header.
A600 and A1200 can simply use a PCMCIA card. The typical choice is ne2000-compatible cards using the cnet.device driver.
I also use cnet.device in my A1200.
As an aside, the ne2000 cards do also work on the Amiga ports of Netbsd and Linux.
You can even get TCP/IP stack and a Web browser running (though things are very slow and many sites do not work).