It's probably most typical to have a server running anyway (that's serving up local pages/content).
However, in addition to the wonderful little EPS1s, lantronix also makes (made) device servers that have mini web server support.
One of the older discontinued models (read: cheap on eBay)was the 'UDS-10/UDS-100' series. You still get the serial port/lan connectivity (although it uses different firmware and lantronix configuration software). This has some html web server support, and you can write custom java apps to run on it. Lantronix even had a software development kit for it.
I stumbled on to the capabilities while perusing for other lantronix device servers (during an ebay drought of EPS1). I picked one of these up in my last big batch buy of EPS1s. I have played with it only a little, as I've been busy with other things, and haven't touched it in several months. But I believe I did get LAN to serial port working.
Of course, Lantronix has modern versions with updated functionality etc, but they are more pricey.