Actually trying to directly make HTTP requests from the Puck via some means is going to be difficult, but I feel like you've got a few different options:
Use the code from https://www.espruino.com/BLE+Advertising so you're using Bluetooth Advertising from the Puck. This isn't quite as flexible (it's not two-way to the Puck) but it is super robust and easy (it's what the EspruinoHub stuff used by default too).
Make the Puck a Bluetooth HID keyboard (https://www.espruino.com/BLE+Keyboard) - you can then just pair it with the PC/Raspberry Pi and make it 'fake' different keypresses to do what you want. I've seen Pucks used this way for accessibility a few times.
Espruino is a JavaScript interpreter for low-power Microcontrollers. This site is both a support community for Espruino and a place to share what you are working on.
Hi! This sounds like a great project!
With Bleak, did you try the code on https://www.espruino.com/Interfacing#bluetooth-le ? What error messages do you get?
Actually trying to directly make HTTP requests from the Puck via some means is going to be difficult, but I feel like you've got a few different options: