If the device used X (some, not all) here is an x86 emulator in the X server, AFAIK it's there specifically for running BIOS code. I've never tried using it myself but I believe it's supposed to work.
In the specific case of ATI products, the VBIOS code is written in an interpreted bytecode language anyways - we released interpreter code in C which compiles down to run on whatever processor the drivers are built for.
In either of the above cases the video would not get initialized until the driver started up, unless the ARM device's firmware included an x86 emulator to handle common PCI video cards in the slot. If the plug-in card were targetted at an ARM device, then the AtomBIOS interpreter could be compiled down to ARM code rather than x86 code.
How are the video cards initialized without an x86 processor to run the video BIOS POST routine?
Depends on the card. Some cards (AMD radeons for example) can be posted by parsing initialization tables in the rom on the card. The kms radeon driver already does this. If that's not an option, you can use something like x86emu to post the card.