I got an answer from Canon on Tuesday:
Thank you for your query regarding the Firmware update for your EOS 5D.
As the versions available for download are only compatible with Windows and Mac, we could produce a zipped up version of the .FIR file so that you can update your firmware on your Linux computer.
Please note that whilst we try our best to ensure the file we provide you with is virus free, you must install the firmware entirely at your own risk.
Could you please also confirm that you have an EOS 5D.
Some progress then. However, they ask me some more questions, to which I can give the answers but you can't reply directly to their support email address, you have to go through the standard form on the web site again. I did that and asked the following question:
Yes, I can confirm that I have an EOS 5D and I am happy to update the firmware at my own risks. Have Canon got any plans to offer a zip version of the firmware in the future in order to support all their customers equally?
So now I'm waiting for the next answer. This whole malarkey of going through the web form every time means that there are potentially 2 to 3 business days delay for the answer to each new question even if it relates to the same problem, which can potentially drag things out a lot, especially considering the web form is limited to 4000 characters!