I believe this claim may have originated from (or at a minimum was perpetuated by) the book Kitchen Confidential. Therein, then New York chef Anthony Bourdain admitted that most fish suppliers in NYC are closed on Mondays thereby forcing restaurants to use older fish. If you accept Mr. Bourdain's word, then I think that confirms the claim. That book was written a decade ago, so I am not sure if that is still universally the case in NYC.
The Fulton Fish Market, which is the largest in New York (if not the USA), is currently closed on Saturdays and Sundays. When they are open, however, the fish is very fresh:
Fish is flown and trucked in from locations all over the united states, and flown in from points all around the world.
I can't find a citation supporting this, but I do know from some chef friends that fish from Tokyo's Tsukiji fish market is flown in to Fulton on a regular basis. It is well known that Tsukiji's fish are very fresh, too:
Most of the fresh/live seafoods will be imported by the [sic] air freight to the Narita international airport or Kansai new international airport under the consignments sales agreement.
Customs clearance at the airport will be done within 2-3 hours after the cargo arrival and immediately to truck to the Tsukiji fish market to sell...
One of the other huge independent suppliers of high-end seafood to the mid-Atlantic region of the US is Samuels & Son. Based in Philadelphia, they deliver to sushi restaurants from New York to Washington, DC, and are capable of delivering frozen, fresh, and even live seafood to almost anywhere in the mid-Atlantic within 24 hours, any day of the week.
Specifically related to sushi, however, one should note that many of the popular fish served in sushi restaurants, e.g., large fish like tuna, are delivered frozen and will therefore not be as affected by delivery dates.
Therefore, it really depends on from where the sushi restaurant is procuring its fish. But to answer your question, there are still suppliers who do not make deliveries on every day of the week. However, there are some suppliers that make deliveries 24/7, so the claim is not by any means a hard and fast rule.
Anecdotally, my local fishmonger (who is not located in NYC) admittedly is only able to receive deliveries of fresh fish every day except Wednesdays, so everything that is sold on Wednesdays is at least a day old.