I finally got an Intel Pro/100 card working on a FIC VIP-486-IO board, but it was a real pain in the butt. I am documenting the process in case it helps anyone.
Card: Compaq NC3120 (this is just a Compaq branded 8255 card that shows up as "Intel 8255x-based PCI Ethernet Adapter (10/100)", same as Intel Pro/100, drivers for Dos/Win 3.1/95/98 from here). Windows 98 and newer already come with drivers for these cards.
OS: Windows 95
Couldn't install Windows 95 with the card inserted, it kept freezing at the hardware detection step. Installed it later, but after a reboot, I kept getting a Windows popup "The network card xxx is faulty and needs to be reinstalled", and in device manager, it showed up with a yellow question mark. There weren't any resource conflicts though.
I figured it was at least half-working, as the Intel diagnostics tool could communicate with the card and detect the cable connection, and at one point, I could even see the PC's name come up in the router. But the tool kept saying that it couldn't properly load the network drivers or obtain an IP address.
When I tested the card on another board, I got exactly the same errors, but re-seating the card to another PCI slot immediately solved the problem, so it had to be some sort of IRQ issue. After an odyssey of fiddling with different PCI slots and BIOS settings, this is what finally worked:
Network card in PCI slot 1, graphics card in PCI slot 2, no other cards installed, with these BIOS settings:
Tested this with IRQ 5 and IRQ 9 for the network card. Note the "INT A using IRQ 9" setting below - It needs to be set to the network card's IRQ or you will get the "faulty network card" error in Windows. The graphics card somehow didn't need that setting. "CPU to PCI write buffer" needed to be disabled or the computer freezes when doing any network related stuff. Also saw a page in the motherboard's manual about "PCI IRQ pull up/down" that says a corresponding jumper needs to be set for each IRQ that's occupied by a PCI card, but that turned out to be a red herring - it never had any effect on any PCI card I have tested.
Also worth noting, the PCI graphics card refused to work with any other setting than IRQ 11 - Windows just boots to a black screen.
Edit: Out of curiosity, just put the network card from slot 1 to slot 3 while keeping IRQ 9, to see if the slot number really makes a difference. Now I get glitchy graphics in Windows, and device manager says there's a resource conflict with the graphics card, which is still set to IRQ 11. It won't even tell me where the conflict occurs, the resources tab just says "unable to load resources". This is so frustrating because there's seemingly no logic behind it.
Could this have something to do with bus mastering? I remember reading that some PCI cards require bus mastering to work properly, and that the motherboard chipset needs to support it, and sometimes only has it enabled for the first and second slot. This could explain why the network card didn't work in the third slot on both boards that I tested.