Oh wouldn't it be loverly
Speaking as someone who works in the legal sector, I would absolutely love it if this came to pass. I don't know a single lawyer, court reporter or judge who doesn't think that an administrative overhaul is massively overdue. If you don't work with the system you won't realise just how much paperwork there is; and considering the scale of the operation it's incredible that more records aren't just lost - other public sector institutions (NHS, I'm looking at you) are terrible by comparison (though of course they may be working under different stresses). It's going to be really, really hard to do, though, and a big part of the problem is the kind of institutional cynicism and apathy that sinks in when looking at really big problems.
If this is going to work, the two projects - the online court and the paperless court - must be separated. They are both incredibly ambitious and the failure of one should not damn the other.
The paperless court must also start off very small. Pick a single, minor court (small claims would be a great example if you weren't going to shove the entire thing online, but maybe you can use it anyway), talk to its admins, watch it closely, and automate as much of that as possible. If it works, pick another, similar court and do that as well. Build templates and slowly, slowly move it out into other courts; remember that premature optimization is the root of all evil. Once people see how much time it saves they'll come aboard. Do not, for the love of God, try and do the whole system at once. It cannot be done. Having small cells of paperless-ness will of course be less efficient than if the entire system were paperless, but that's just the price you pay for having a working solution.