The first question (for mythmon) is simply whether Normandy can do this, it would need to install the desktop webcompat extension for every desktop user and likewise for mobile. If we were to do this, how would the process be different from today for Tom and the webcompat team?
In short, Normandy can't do this (yet). Normandy can currently only install extensions experimental use cases. In bug 1436111 we're working on adding in the capability to install extensions for permanent, non-experimental use cases. I expect this will be done sometime in the next few months. Once this is finished, I expect that webcompat would be a good use case for the feature.
Further, Normandy does not exist on mobile at all, and can't do anything there. I think it may be possible to enable Normandy on Fennec, but I'm not entirely sure yet. There may be dragons hiding that we don't yet know about. This is not a high priority for my team at this time.
As for how the process would be different, I'm not sure for two reasons: I don't know much about the existing Balrog process, and the equivalent Normandy process doesn't exist. That being said, I expect the process to be about the same, or slightly easier.