Was flooding from shell penetrations a real problem to amphibious tanks in world war 2? (American LVT or Japanese Type 2 Ka-Mi for example). If so, how did the crew cope with it, did they have the necessary equipment?

In the case of the LVT, only if the vehicle ran out of gas or was simply too badly damaged to stay afloat. The vehicles were equipped with engine-driven bilge pumps. The most dramatic examples of this were during the battle for Tarawa in November 1943, when fifteen damaged LVTs sunk on the first day of the battle after attempting to make a return trip to their mother ships, crossing back over the reef, and immediately encountering deep water, and at Roi-Namur (the northern phase of the battle of Kawajalein) in February 1944, where twenty-three LVT-2s had to be abandoned and eventually sunk due to taking on water when timetables were disrupted and craft carrying additional fuel failed to appear. Once combat experience was gained, it was a common practice to carry a number of tapered wooden plugs or rags in order to perform on-the-spot "repairs" of holes from bullets or shell fragments. A limited number of LVTs carried hand-operated bilge pumps in case the engine-driven pumps failed.

Sources:

Bailey, Alfred D. Alligators, Buffaloes, and Bushmasters: The History of the Development of the LVT Through World War II. Washington, D.C.: History and Museums Division, Headquarters, U.S. Marine Corps, 1986.

Camp Colt to Desert Storm: The History of U.S. Armored Forces. Edited by George F. Hofmann and Donn A. Starry. Lexington: University Press of Kentucky, 2012.

Croizat, Victor. Across the Reef: The Amphibious Tracked Vehicle at War. London: Blandford Press, 1989.