Unfortunately due to your employer's data privacy your work computer won't be able to install software other then what's approved by the IT department. There is no way around this security block that they set up for their Network protection.

I am not trying to install software. I am trying to access the data that is stored in Firefox's Sync service.

It still has to download it to the computer to use the sync data as it has to write that data to Firefox locally to use it. That is considered installing software outside of the company data privacy protection.

I do not need to download anything with any software. I just need to access the data that is stored. I am assuming there should be some website somewhere that Mozilla would have made available so that you can just log in from your web browser and see the stored data. After all bookmarks are commonly stored in plain text HTML so its already formatted to be viewed in a web browser.

I don't think you could use Sync without Firefox.

You can only use external add-on to sync bookmarks through Google Drive. Check out https://floccus.org/download/

Thanks, I am trying out this Floccus thing, will have it sync to some GitHub repo that I can just view in the web browser.