I don't think you have thought this whole thing through. I have done some considerable lightweight thread apps with Greenlets created from the Gevent framework. As long as you allow control to switch between Greenlets with appropriate sleep's or switch's -- everything tends to work fine. Rather than blocking or waiting for a reply, it is recommended that the wait or block timeout, raise and except and then sleep (in except part of your code) and then loop again - otherwise you will not switch Greenlets readily.
Also, take care to join and/or kill all Greenlets, since you could end up with zombies that cause copious effects that you do not want.
However, I would not recommend this for your application. Rather, one of the following Websockets extensions that use Gevent... See this link
Websockets in Flask
and this link
https://www.shanelynn.ie/asynchronous-updates-to-a-webpage-with-flask-and-socket-io/
I have implemented a very nice app with Flask-SocketIO
https://flask-socketio.readthedocs.io/en/latest/
It runs through Gunicorn with Nginx very nicely from a Docker container. The SocketIO interfaces very nicely with Javascript on the client side.
(Be careful on the webscraping - use something like Scrapy with the appropriate ethical scraping enabled)