I just filed my HHT claim with Sirva... The process was simple, well laid-out, and required exactly zero paper. If our relocation people can figure this stuff out, there is no reason the CAF can't, apart from dinosaurs making decisions based on their personal preferences.
It was a massive fight to get my occupation to switch to electronic weather records from the old paper ones we had been using since the mid-90's. What finally forced the organization to change was Environment Canada finally running out of the forms, and the cost to have them printed ourselves. I fear that if it happened under the current government, we would have just paid to print more rather than adapt to the modern world.