Thinking though which of my vintage computer collection might be able to do this..
The first problem is that almost all of them predate the Internet. This means TCP/IP stacks and web browsers would need to be add-on software modules. Several are so old they predate most networking technologies. So here is the systems in reverse date order...
The easiest one would probably be the NeXT Cube, as this has most of the components in place (as well as being where the web was invented): this is 1990 onward (not sure what year mine is). I'd have to get it down from the attic, set it up and figure out how to break into it (I bought it as-is with an existing OS setup but no usernames or passwords for the system).
I have several old Macs - a couple of 512Ks, one 128K and an SE. A quick Google search suggests this would be possible if I had another Mac with both AppleTalk and ethernet (which I don't). But system 7 would run on the SE so in theory that might work. An alternative might be to plug a modem into it via serial and use dial-up networking to my network via the phone system (old-skool!). Don't have any Mac modems though. It would have to be the SE - I'm not loading it from FDD!
Going back a bit further I've got two (or possibly three?) IBM PCs: one 5150 and one/two XTs (I think?). One of these might have an ethernet card in them, or I could plug a modem into the serial port. I think I have a couple of old modems about. For the software MS Windows isn't an option (except Windows 1.x) - these are 8086 and 8088 CPUs so it would need to be a DOS-based web browser and TCP/IP stack (they do exist). Tricky but not impossible.
Several more eclectic/British models I don't think could do it are the Dragon 32, Sharp MZ80B and MZ80K, the Video Genie and a Research Machines RM80Z, so we'll skip those. I did think about the Sinclair/Timex ZX80, but I'm not sure how the networking would be achieved. Writing a "web browser" could be done in the ZX80's BASIC interpreter, but as it only has 1KB of RAM I'm not sure it would be achievable.
Next up are two Commodore PETs, a 2001 and a 3032. These have an IEEE 488 interface so something may be possible to hook into that, but I've no idea what.
My TRS-80 Model I does not have a serial port but I think I have an expansion interface for this, which may do. So we'd plug that into a modem, and then use this project as a base to network it.
Finally it's almost certain that my Apple ][e would be able to do this given the number of mods, boards and hobbyist stuff out there.