What's the best practice for interacting with a browser app (say, HTML5), connecting to a Teensy's running instance?

I'd like to send and receive data between the browser app and the Teensy app for:
- viewing and changing the Teensy App's config
- harvest logging from the Teensy

I've not found an answer in my forum searches, though I don't know that I have the right search terms.

I've looked a bit into the Web MIDI API - which might work in Chrome (only?) for exchanging MIDI messages over USB MIDI.
I've found a Web MIDI sample app here which sees my Teensy MIDI device, and receives it messages.

Are there other standard ways that are commonly used for these purposes listed above?