Bangle.js emulator: Module http not found

I've thought that, as soon as the device is connected/paired through Bluetooth, that the puck/bangle would have access to the Internet

No, I'm afraid that's not the case. There is no proper support for HTTP or Networking in Bluetooth LE, so there's nothing that 'just works'.

There is the Bluetooth LE HTTP proxy that you can use with EspruinoHub but it's extremely limited in terms of the data that you can transfer: http://www.espruino.com/BLE+HTTP+Proxy

How would I bridge the puck.js through an android device to the internet?

Web Bluetooth would be the best bet as you suggest. It's easy to develop and multiplatform, but you could write an Android App to handle it too if you want something always-on.

Basically there's no easy way to do an HTTP request direct from Puck.js. The best way is to do the HTTP request on your 'bridge' device and just send the data you need back to the Puck.js over Bluetooth.

It's a pain, but Bluetooth LE just doesn't have support for networking in it - it's not what it's meant for.

when I run the first example code from here: http://www.espruino.com/Web+Bluetooth
on my webserver

Are you using HTTPS? Web Bluetooth only gets enabled for HTTPS sites, not HTTP. (ahh - I just read at the bottom that you figured this out).