Some food companies are refusing to make deliveries unless they are paid upfront and there are reports of empty store shelves as panicked consumers try to buy food. Also, key farm inputs such as fertilizers, pesticides and fuel also are usually imported and will require payment in advance in hard currency that is no longer available.

"There's a lot of speculation and even more tension," said George Frangistas, president of Incofruit-Hellas, the trade association of Greek export enterprises for fruits, vegetables and juice. "Clearly with closed banks and political pressure boiling over normal trade cannot go on."

Greek's agriculture sector was already facing a challenging 2015 before the economic turmoil this week, but now there's fear the uncertainty could disrupt the food and wine trade.

"Imports (have) already ground to an immediate stop and exports are limited to the available transport currently in Greece," said Frangistas, who also serves as managing director of exporter Gefra.

Read MoreBlame game: Greece in deadlock ahead of vote



Even before Greece headed towards default on its IMF loans on Tuesday there were troubles brewing on the agri-food front. Last year, Greece's ag sector got hit after Russia banned EU food imports. Russia was once responsible for importing more than half of the Greek peach crop and most of its strawberries, and the Greek kiwi fruit market was growing, too.

Greece's total ag exports last year totaled about 5.5 billion euros, of which fruit and vegetables represented about a third, according to Incofruit. In the U.S. market, top Greek imports include olives, olive oil, honey and wine.

Ted Diamantis, president of Diamond Importers and Naturally Greek, termed the current situation "frustrating." The Chicago-area businessman imports wine and food products from Greece and just returned from a business trip to Athens. He said businesses have been told they should be careful about transferring money to Greece because of a perceived risk of the IMF seizing funds.

Read MoreGreek crisis: Tsipras and creditors in standoff



Agriculture, including crops, dairy, wine, fishing and forestry, makes up about 3.5 percent of Greece's GDP and represents about 13 percent of the labor force. Olive oil represented nearly 9 percent of Greece's total agricultural output last year, according to Eurostat.

Earlier this year, Costco changed the sourcing of its Kirkland brand olive oil from Italy to Greece. That followed a weak 2014-15 harvest in Italy that caused prices to soar for Italian olive oil. Costco didn't respond when contacted for comment.

About 25 percent of the Greek olives get shipped to the U.S. market, and one of the most popular table varieties is the Kalamata olive used in salads and named for a city in southern Greece.

Meanwhile, Greek wines are enjoying growing demand despite increasing worldwide competition. Greek wine is shipped to more than 35 countries around the globe but three countries—Germany, France and the U.S.—represent about 66 percent of the total volume exported.

"We have not experienced any issues in receiving product from Greece," said Andrea Englisis, president of Athenee Importers in New York. "We have more orders that are set to leave this week and have been advised that there will be no interruptions in service."

Diamantis believes the Greek wine industry should be able to weather the current crisis because there's strong demand at home for the Greek wine as well as in the global marketplace. "There's a silver lining to all the turmoil," he said, adding that consumption of Greek wines is trending higher at home as pride builds during the crisis and people seek out local wines to drink.