Here are some
possible solutions (haven't tried them myself) which don't require a network card so they might be a bit easier and lighter on memory usage, although much slower. I guess using RS-232 serial is also a bit more period correct for the experience many of us had, but these also might be harder to get working since you won't find so many explanations online about how to do this stuff.
I assume you could get Trumpet Winsock to talk over a null modem cable to any SLIP or PPP server. My Linux system has an
slattach
program which can apparently be used to start SLIP or PPP on a serial device. I seem to remember a long time ago getting some old Cisco router to talk SLIP or PPP on one of its RS-232 interfaces. I also seem to recall it wasn't too hard getting SLIP working on Linux manually in the distant past. Just remember that SLIP has no IP autoconfiguration like PPP, so if you use SLIP you need to provide compatible IP settings on both ends of the connection.
twinsock might be another option. It might be simpler because it just uses a userspace program on the Internet-facing/host system rather than something like
slattach
which is relying on some kernel support and hence might have permissions issues. The problem with twinsock is it's actually from the Windows 3.1 era, so it might be hard to build the host-side program on a modern system. You might enjoy running it on an emulated retro system though, as neozeed did. Unfortunately in this case he didn't just provide a disk image of his 386BSD system he used!