Ted Diamantis, an importer of Greek wines who is based in Chicago, has been helping his suppliers stock up on bottles, labels and printing ink. The barrels, though, have him worried.

In two or three weeks, some of Greece’s winemaking regions will begin their annual grape harvest. The wineries Mr. Diamantis buys from age their wine in barrels from Italy and France, but Greece’s capital controls make it difficult for them to send money out of the country to pay for the barrels they need for this season. No barrels means no wine for Mr. Diamantis.

“Without the ability to access your capital, you can’t buy anything,” said Mr. Diamantis, who is in Greece meeting with his business partners. “The marketplace is frozen.”

American companies that import Greek products have been battling uncertainty for the last month. Export shipments are still leaving the country, but Greece’s debt crisis is disrupting its supply chain in both significant and subtle ways, threatening the livelihoods of Greek merchants and their distant trading partners.