Every now and then Matthew Frank, whose company makes Tovolo-brand ice-cube molds and trays, will slow down shipments of his kitchenware to Amazon.com Inc. and then conduct test buys as the Internet retailer's stock runs low.

What he finds after cutting back the flow of his products: Knockoff products rise to the top, said Mr. Frank, chief executive of ICI USA LLC, which owns Tovolo.

The...