EDI is more standardized data formats than API (though the fact that it often standardizes request and response formats and the relation between them makes it close to an API in some requests), and EDI goes a lot beyond manufacturing and supply chain management.
(In US healthcare financial transactions, specifically, the combination of EDI transaction specifications + standard operating rules mandated under the HIPAA is, basically, a standardized API, if not called that.)
Given the Expense + complexity of EDI, in all cases we've found it more cost effective to integrate with all the alternate Web-based import/export tools provided by systems like VendorNet, or whatever other crazy systems our customers use.
If our volume gets bigger, we'll probably have to revisit that position, but it's served us very well for now.
As far as retail stores are concerned, they make an API call to a supplier, and product shows up.
Granted, it's a pretty convoluted API, but I don't think it's too difficult to imagine what a supply chain algorithm would look like. Just lots of exception handling + long timeouts...