You could expose your library with a plex service. Now you can play it anywhere using any plex client. You could host a web client anywhere and have it play files from your local plex service (as long as the plex service is exposed to the internet)
All of that is very achievable, except the "web app" part. I think it might be about time for me to just give up on that pipe dream and start building an Electron and/or React Native app instead.