If a Nintendo Wii counts as retro, there's at least one person hosting a website using one: https://infosec.exchange/@alexhaydock/114377262481451962
But that was (incredibly!) able to run a prebuilt version of NetBSD, which had drivers for the official Wii USB LAN adapter. If instead there's a piece of hardware that can boot an OS but doesn't have a network interface device, then SLIP could be useful. Here's a vape hosting a website: https://bogdanthegeek.github.io/blog/projects/vapeserver/
And if still, said retro device has no concept of IP or networking, can it at least speak serial over a 300 baud modem?