So we start with the time line. When did Japan enter WWII? Not very simple because Japan had already initiated wars with China in 1931 and in 1937 with whom we were allied at the time. But Japan didn't become allied with Germany until September of 1940. No conflict with the story yet.
If you read up in old chemistry texts of the 1930s, you'll see that the primary producers of shellac in that era were India, Indo-China, Siam and the Federated Malay States. (Some sources also list Vietnam) Today we call them India, China, Thailand and Malaysia. Most of them still are as well. We just don't make records with it any more; instead we coat Junior Mints with it. (They call it confectioners glaze.) [source] But what doesn't ring true is one single ship pinching the supply enough to stop the presses everywhere. Because the harvest of any natural product is both seasonal and volatile, smart companies stock up on such mission-critical raw materials. How big was that payload?
What the story skips is that even before 1943 we were rationing shellac. In 1942, the War Production Board cut domestic shellac consumption by 70 percent. It limited them to the raw materials to press about 50 million records. Then they also froze prices at 1941 levels, there would be no price gouging. The only silver lining here was that labels could still press from their shellac stock, and sell the records already in inventory. In June of 1943 Billboard published an article titled "New Ruling Due July 1 On Shellac Allowance." It read in part:
"Many record company executives... consider it likely that the War Production Board will cut them off completely from any virgin shellac use. they base their view on the fact that the country's current shellac stocks fall short about 2,000,000 pounds the amount the army and navy require in the next 12 months... Some shellac has been arriving from India but the amount shipped has been small. India's shellac has been gathered and it amounts to 45,000,000 pounds."Like many other goods, shellac has multiple purposes, and some are for the military. At the time, all electrical wire was wrapped in cloth and waterproofed with shellac. That sounds a tad important. In June of 1942 the War Production Board cut them back to 30% of their 1941 tonnage per month, then by August of 1942 they were cut back to 15%. It would dip to as low as 10% at times. This led to a couple reactions: stretching the virgin supply through use of lower grades, lamination, fiber cores, but also shellac recycling. In 1942 and 1943, some record labels began to hold "Shellac drives," with live entertainment where consumers turned in old records to be recycled into new 78s. Wartime rationing was finally rescinded on April 1st 1944.
While over 375 American ships were sunk in WWII just in 1943, [source] I can find no record of some massive shellac shipment that would bring the whole industry to it's knees. It was already functioning on a tiny fraction of it's normal supply, the contents of a single shipment would have meant only a trivial change in availability. When I have found a manifest listing payloads the shellac is usually listed as less than one ton. It's entirely likely that some supply ships were sunk. But while the supply-side bottle neck was very real, the single-ship theory is clearly bunkum.