Getting usage data from Twine isn't too hard, you could collect whatever information you want and pretty easily pack it up as JSON or similar. From there though, things get a little complicated, since you'll need some way to send that data, and you'll need some sort of system in place that receives and processes that data, too. Traditionally, you would use a server and a database program of some kind for this. Even assuming you have access to a server and the ability to configure it and program it, I think that that level of programming goes quite a bit beyond the sort of help you tend to find here, and certainly goes well beyond my own abilities. If this is the approach you want and is feasible for you, I suggest looking elsewhere for those sort of technical questions, like Stack Overflow.
If that's not in your wheelhouse, you could instead have twine output the JSON data representing the student's usage information and have the students themselves send it directly to you via email or similar, and then unpack and parse that data yourself. The problem here, though, is pretty obvious. A single error in the copy/paste means a student's work could be lost, and if you have a lot of students, that could be a lot of work. You'll also likely want to encrypt this data, at least a little bit, to discourage cheating (assuming its an issue), and that makes things more error prone on the copy/paste side since it'll be harder for all parties to determine if the data is complete until you decrypt it. This is more realistically achievable assuming you can't or don't want to go the server route, though it still might not exactly be a walk in the park. If you want to try to implement something like this, let me know which story format (and version) you're using and I can write up some example code to help get you started. That said, it's not really an elegant solution.
Someone else might have a better idea if neither of these solutions are what you're looking for.