SEATTLE—Fans of farmers' markets don't always agree on the fine points of what defines the folksy bazaars, but they concur on what farmers' markets aren't: chain grocery stores selling fruits and vegetables on their supermarket doorsteps.

Farmers and their supporters have spent several decades building "farmers' market" into a brand that signifies something specific to consumers, namely, locally grown produce fresh off the farm. Now, to the dismay of farmers' market representatives,...