During WW2, a German invasion of the Isle of Wight was not carried out for several key reasons. The main one was the failure of their overall plan for the invasion of Britain, Operation Sea Lion (Unlike the Channel Islands, which were invaded in June-July 1940, the Isle of Wight is so close to the British mainland that it can effectively be considered part of it). This went through various iterations, several of which involved landings on the island. However, the plan failed, primarily due to the strong opposition the Germans faced from the Royal Navy and Royal Air Force, as well as German inexperience with amphibious operations. I have covered this plan and its failure in more detail here and here.

The island was not especially heavily militarised. The main military presence were the coastal defence batteries protecting the entrances to the Solent, as well as the RAF radar stations. The former included 9.2in gun batteries at the Needles and Culver Down, as well as 6in guns at Bouldnor and Nodes Point. The RAF had Chain Home radar stations at Ventnor and St Lawrence, as well as four Chain Home Low stations. In 1940, the defence of the island was the responsibility of the Army's 12th Infantry Brigade. This was, in early 1941, replaced by the 214th Brigade, along with troops of the 19th Battalion Hampshire Home Guard. During this period, several lines of pill-boxes were constructed, including a stop line following the route of the river Yar. However, this was well after the greatest danger of invasion had passed.

All told, the main defence of the island was not military, but geographical. The best beaches for a landing, and the major ports, lie on the north coast of the island. Much of the south side of the island was protected by cliffs, with the few beaches covered by coastal defences. Supplying troops over a beach was difficult without artificial harbours like the Allied Mulberry, to which the Germans had no counterpart. As such, to land and supply troops, the Kriegsmarine would have had to enter the Solent. This was covered by shore batteries at both ends (on the Isle and on the mainland), as well as the sea forts protecting the eastern entrance. Entering the Solent also put the German supply convoys within easy reach of RN ships and submarines from Portsmouth - a threat the heavily outnumbered Kriegsmarine couldn't easily brush off. If the Germans landed in the south without acting in the Solent, the British troops on the island could easily be resupplied through the harbours on the north coast. Meanwhile, their German opposition would have to make do with what the Kriegsmarine could land on the beaches. A further component to the defences lay offshore. The island was very close to the mainland, and to the RN's main anchorages. Batteries on the mainland or RN ships in the Solent could easily provide shore bombardment in support of troops ashore. The battleship Revenge, based at Spithead from late September 1940, could hit most of the island from the anchorage, without even having to weigh anchor. Finally, the island lay at one of the wider points of the Channel. The Kriegsmarine was uncertain of its ability to carry out an invasion at the Channel's narrowest point, the Dover straits. Increasing the distance to travel meant that the Kriegsmarine would have to provide more escorts, and more transports would be needed due to the increased travel time.

A further point dissuading the Germans from landing on the Isle of Wight was that it was of questionable strategic significance. While it could, if completely captured, effectively neutralise Portsmouth and Southampton as naval bases, these were not the only RN bases on the South Coast. Plymouth and Dover were just a short steam from Portsmouth, and could replicate most of the functions of Portsmouth. The island had only a single operable airfield, at Somerton (all others had been closed and obstructed). This had a grass field, so was only really suitable for the lighter aircraft in the Luftwaffe's inventory. Given that the island could easily be brought under artillery bombardment, this airfield was thought to be not especially useful. The island had little industry, while its farms were mostly needed to supply the needs of the population. What could be looted would need to be shipped across the Channel in the face of severe RN and RAF raids.