The trivial non-tracking implementation would be for edge to install the entire database of products and prices to your local device.
Whether that is practical or not depends on just how large the database is. But even if the database were to be too large, it'd still allow us to reduce the question to one of how to download the database incrementally to the device without leaking information, which is a solved problem. (E.g.the Safe Browsing algorithm.)
The trivial non-tracking implementation would be checking if the domain of a requested page matches a certain list of known e-commerce sites. If so, then on load, query the other known e-commerce sites through a proxy to see if there's a cheaper price. No tracking necessary.
That would be both highly non-trivial and non-private.
For the non-trivial part, the approach would be hell to scale to any significant number of e-commerce sites. You can't have the client connect to tens of thousands of sites on a page load. And how do you reliably find the matching product pages on the other sites? It's easy if you can crawl the sites, but extremely hard if you tried to do it with point lookups.
As for the privacy, who operates the proxy? Why is giving them both your IP address and a stream of all commercially interesting web pages you visit not a tracking concern?
Whether that is practical or not depends on just how large the database is. But even if the database were to be too large, it'd still allow us to reduce the question to one of how to download the database incrementally to the device without leaking information, which is a solved problem. (E.g.the Safe Browsing algorithm.)