This could indeed be possible. But since you want to save the animation on the client side, you will have to create some custom code (both on server and client) to achieve that.
Visualizer could be a place to start but I’m worried that there will be too much knowledge to grasp before going anywhere.
In some way, it sounds like you are looking for a custom application that would satisfy your user workflow with a fairly limited set of required features. Maybe that workflow could even be achieved purely on the client side with vtk.js and a basic web server.
It could be worth checking with Kitware for a support contract to get you started on your project.
HTH,
Seb