I'm mostly referring to the use of LVTs that were able to better support landing infantry. When reading about D-Day, I continuously hear stories about how troops were towed across the Channel in some landing craft.
I would think, for the largest and most essential naval invasion in history, that the Allies would have relied on LVTs.
The LVT was not used in the majority of European amphibious operations, though it would see some use with British forces during the clearing of the Scheldt Estuary. The main reason for this was geographical. The main advantage of the LVT was its ability to cross shallow offshore obstacles, like reefs or sandbanks. A landing craft would beach on these, leaving the troops with a long wade to the beach. This would leave them exposed to enemy fire. If these obstacles did not exist, though, landing craft had the advantage of being quicker in the water, and carrying more men - the LVT-4 carried 18 men at a top speed of ~5 knots, while the British LCA, one of the slower landing craft, carried a reinforced platoon of 35 men, at a speed of about 7 knots. Landing craft, being made mainly of wood, were also easier to produce in bulk and relied less on strategic materials. As such, it made much more sense to use landing craft where reefs or sandbanks were not present.
In the Pacific, many of the islands that were attacked were surrounded by reefs. To attack one of these islands meant crossing the reef. In the European theatre, though, the Allies had the ability to pick and choose where to attack, thanks to the much greater length of coastline. They could, and did, choose beaches that were free of any obstructions. Given this, they chose to use landing craft to a greater extent. This freed up LVT production for the Pacific, where they were much more useful. However, when time came to clear the Scheldt Estuary (necessary to allow shipping to flow into the port of Antwerp), LVTs were used. This was because the estuary was full of mudbanks, which played to the LVTs strengths.
While the Allies did not use LVTs, they used other amphibious vehicles. DUKW amphibious trucks were commonly used in amphibious operations from Operation Husky, the 1943 attack on Sicily. These were unarmoured vehicles, so were rarely used in the first wave of an assault, but were very useful for ship-to-shore transfer of troops and supplies in later waves. More famous were the DD tanks. These were Sherman tanks modified with canvas flotations screens and a propeller system to drive them through the water. They were first used in the D-Day landings, with moderate success. Launched from too far offshore in the rough seas, many sank, especially off the American beaches. They were more successful during Operation Dragoon and in the Scheldt.