I reached out to Oliver Schmidt, maintainer of Contiki for Apple 2 (among other things), and he suggested the following:
You can't "emulate" an Uthernet card with an RPi but you can -
generally - use the RPi to get online with Contiki.
* Warning *
I haven't tried what I describe below myself. And I have a report that
it doesn't work reliably. And it's quite some effort. Nevertheless...
Instead of using an Ethernet card Contiki can be built to use a
SuperSerialCard to connect to a SLIP server. There's no pre-built
Contiki for SLIP, you have to build it yourself.
Read
https://github.com/contiki-os/contiki/blob/master/cpu/6502/README.md,
especially
"The cpu/6502/ directory is used for targeting 6502-based machines
using the cc65 compiler http://cc65.github.io/cc65/."
and
WITH_SLIP Default: 0 Purpose: Use SLIP (based on RS232 driver coming
with cc65) instead of Ethernet.
additionally read
https://github.com/contiki-os/contiki/blob/master/platform/apple2enh/README.md,
especially
"The 'disk' make goal requires AppleCommander 1.3.5 or later. It is
available at http://applecommander.sourceforge.net/."
You can either build the individual Contiki programs using e.e.g
https://github.com/contiki-os/contiki/blob/master/examples/webbrowser-80col/Makefile
or build them all using
https://github.com/contiki-os/contiki/blob/master/tools/6502/Makefile
Getting the RPi ready is supposed to be easy, e.g.
https://harizanov.com/2013/03/tcpip-over-rfm12b-and-raspberry-pi-as-gateway/
says that something like
sudo modprobe slip
sudo slattach -L -s 38400 -p slip /dev/ttyAMA0
sudo ifconfig sl0 192.168.5.1 dstaddr 192.168.5.2
is enough.
Please note that I don't know more about this then what I wrote above.
Good luck. Oliver
I will update this with my progress doing so, and try to post the rebuilt Contiki binaries somewhere as well. Thanks to Oliver for his assistance.
Update: I was having trouble with the compilation, so Oliver was kind enough to compile for me and send along. Here are the compiled binaries if you are interested.