ESP8266 as WiFi monitor
You're going to face significant challenges with this.

The ESP Arduino API doesn't let you run in promiscuous mode (which you'd need in order to see other people's traffic), and the APIs are all set up for you to create nice well-behaved traffic, not spoofed packets.

I strongly suspect the hardware can do it, but you're going to be making significant changes to the IP stack.  The code is all open, so you're free to dig in and see what you can do (I've done a similar thing myself, as I needed some lower-level information than was made available in the API).

So feasibility is determine by your skill level (C, C++ and IP stack and protocols) and available time.

Something like a Raspberry Pi Zero and a suitable USB wifi dongle would probably be a quicker path at similar cost, as there are Linux tools to do a lot of this stuff.