I want to make it possible for my visitors to search through the products of several online stores. What kind of programming might such a function require? Note that I want to be able to do this without storing all of the stores' products in my database.
you can do this in php quite easily. It basically what's commonly referred to as a web-scraper (scraping other sites for content). You can also do this in a different way if the site in question provides an API (eg:amazon), but again PHP would suffice.