While it is a good idea convenience sake, I think it is a bad idea generally, due to every country having it's own local stores, and only supporting a few select stores for a few selected countries is a bad idea if you want to financially succeed as a small company.
As a solution, I think you should be able to hand away the actual product search for the user, but make it so if you search for a specific product, for instance, headphones, you will see a "product page" listing the product's specifications.
This should make it easy for the customer to make a final decision, and buy the right product they need, at the store they prefer.