+1 to Marc B's comment. If the TOS doesn't explicitly forbid it (and since this would also be considered a crawler), you should see if /robots.txt
disallows you from accessing the product search. If neither forbid you, I would suggest using a browser-based bot to fetch results for you, simply because it sounds more practical and you wouldn't have to deal with cookies.
If you want to make the page requests with PHP, though, I would direct you to HttpRequest. Have a page where you can log into all the sites (using a POST request right on the login scripts), and keep the session cookies returned handy. When you search the product pages, identify what part of the HTML consistently returns the list of products after it (a regex may be helpful), and create an algorithm (which should be different for every website you want to scrape) that returns information about the product. Then compare the results!