That's a neat idea. I imagine a service worker could also be used to do checks in the background, and unless SW wants to ban every IP that hits their site.. :-D

Amazon mechanical turk?

Using that you could not only have real people doing this work but also support (for low cost) people in developing nations who need money to survive.

Because of cookie policies they'd need to clear their cookies every time they query.

You could also after a time keep stats to determine when the prices are most likely to go down, and then only get the user's attention at those intervals.