There's certainly a vulnerability:
Here’s how to shut down the internet: Snip undersea fiber-optic cables
Undersea cables conduct nearly 97 percent of all global communications, and every day an estimated $10 trillion in financial transfers and vast amounts of data pass through the seabed routes. Satellites, once crucial but now limited in speed and bandwidth, handle only a tiny percentage of global communications.
For a historical perspective, see:
In the early hours of 5 August 1914, only a few hours after war was declared, Britain carried out something that seemed to be minor, but was actually vital. A British cable ship severed five German overseas underwater cables, which passed from Emden through the English Channel to Vigo, Tenerife, the Azores and the USA. This cut direct German communications to outside Europe, most significantly to the United States. The British could now intercept German signals to their embassies. They were sent in code, but British codebreakers were eventually able to read them. Most significantly, Britain intercepted the Zimmerman Telegram, sent to the German Ambassador to Mexico. If the USA went to war with Germany, he was to offer the Mexicans an alliance with the promise that they would receive Texas, New Mexico and Arizona. Its revelation helped to push the USA into war with Germany...
These operations, carried out in July 1945, were intended to cut the undersea telephone cables connecting Singapore, Saigon, Hong Kong and Tokyo. The intention was to oblige the Japanese to use radio and render themselves open to message interception.
Operation Sabre was directed at the Hong Kong to Saigon telephone cable, and carried out by XE4, which was towed to within 40 miles (64 km) of the Mekong Delta by the submarine HMS Spearhead, where she looked for the two telephone cables by using a towed grapnel. She eventually snagged the first cable, and managed to haul it about 10 feet (3.0 m) off the seabed. XE4's diver, Sub-Lieutenant K.M. Briggs, used the net/cable cutter to sever it. The second cable was soon found as well, and was severed by the second diver, Sub-Lieutenant A. Bergius. Two divers were carried due to the operating rule that a diver should not spend more than 20 minutes in depths over 33 feet (10 m) and no more than 10 minutes over 40 feet (12 m). XE4 and Spearhead returned to Labuan on 3 August 1945.
Operation Foil was directed at the Hong Kong to Singapore telephone cable and carried out by XE5 against the Hong Kong end of the cable, after being towed into position by the submarine HMS Selene. Operating close inshore near to Lamma Island, working conditions were poor, XE5's divers having to work in thick mud under the constant threat of oxygen poisoning. Despite repeated attempts it was not completely certain that the cable had in fact been severed, and it was not until after the Japanese surrender that it was confirmed that XE5 had succeeded in doing so. XE5 and Selene returned to Subic Bay on 6 August 1945...
More recently:
TRIPOLI (Reuters) - Libya’s rebel-held city of Misrata won no respite from two months of bitter siege as Muammar Gaddafi’s forces bombarded the city and battled rebel fighters, despite pulling out of the city centre...
Late on Monday, the “crusader aggressors” bombed civilian and military sites in Bir al Ghanam, 100 km (60 miles) south of Tripoli, and the Ayn Zara area of the capital, causing casualties, Libyan television said, without giving details. A Reuters correspondent heard explosions in Tripoli. The report said foreign ships had also attacked and severed the al-Alyaf cable off Libya’s coast, cutting communications to the towns of Sirte, Ras Lanuf and Brega...