On a brisk fall morning, with the dew still fresh on the grass, volunteers in their old T-shirts and blue jeans listen as Tracy Blackmer shared instructions for their day on the farm.

As soon as he walked them out to the garden, the volunteers got to work picking. They quickly filled several plastic crates with an assortment of peppers in pop-art colors, a bright contrast to the golden corn field behind them.

As of mid-September, the Black­mer family had grown more than...