you'll want to find a way to get the extension pciutils.tcz onto your boot drive which can help identify your ethernet hardware (unless anyone knows of a method TC already has in its core image to identify make/model?)
The list of PCI device descriptions (pci.ids.gz) comes from the pciutils.tcz extension, so there isn't alternative that works as easily as lspci. However it's possible to find vendor and device IDs by looking in /sys/bus/pci/devices then look them up manually.
For example, lspci returns this for my Ethernet interface (my particular model of which doesn't work at all with the Linux driver in TC11 and TC12 and needs the r8168-KERNEL.tcz extension with the official driver):
01:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller (rev 0c)
Without using lspci, I can find the vendor and device IDs as follows:
$ cat /sys/bus/pci/devices/0000:01:00.0/vendor /sys/bus/pci/devices/0000:01:00.0/device
0x10ec
0x8168
Now I can look up the vendor ID at the
web interface to the PCI ID Repository which pci.ids.gz is sourced from (also more up to date than the 2017 copy in the pciutils.tcz extension). Entering "10ec" into the box at the bottom of the page and pressing "Jump" brings up the
Realtek vendor section, then entering "8168" and pressing "Jump" at the bottom of that page brings up the
device page, matching what lspci reported.
/sys/bus/pci/devices/0000:01:00.0/revision (0x0c) also shows that it is actually model 8168G, based on the comments on that device page.
Note that you'll have to do this for all the devices in /sys/bus/pci/devices, which might be a bit laborious. I think the Ethernet interface is most likely to be towards the end of the PCI IDs listed though, so start from the end of the list and work back.