Modified from an earlier answer

The contribution of the CCP to the war against Japan is a highly contentious and politicized issue whose intellectual roots stem back to both the Sino-Japanese War and the aftermath of the CCP's victory over the KMT. The Nationalists popularized a claim during the war that the CCP was devoting only a fraction of its resources towards fighting the Japanese and was building a powerbase to attack the KMT after the war. Conversely, Mao conducted a public relations drive, including the famous Dixie Mission to Yan'an, that argued the CCP had a wide-scale and determined guerrilla campaign against the Japanese, in contrast to the lackadaisical and corrupt KMT. The swift and unexpected victory of the CCP in the Civil War gave currency to the CCP's wartime line. The negative opinions of Chiang Kai-Shek held by many American officers and would reenforce the CCP's contention that the KMT squandered whatever influence it had with the Chinese people because of the Nationalists' rampant corruption and incompetence. Subsequent academics in the 1960s and 1970s like Chalmers Johnson and Mark Selden would further argue that the needs of guerrilla warfare created a large-scale peasant force where nationalism coupled with land-reform had transformed the CCP into a mass movement led by Mao.

Now each of these broad interpretations of Chinese history has their partisans, current research into the Sino-Japanese War like Mitter's is presenting a more differentiated and less politicized picture of events. One of the basic problems of the CCP perfidy vs. KMT corruption debate is that they are predicated upon the notion that China in the 1930s could have fought and sustained a modern war against an industrial power like Japan. This is built on a number of assumptions about China's actual strength and ignores the real limits faced by the KMT. The NRA's battles around Shanghai in 1937 is a case which has benefited greatly from scholarly reevaluation. Chiang deployed many of his best troops to defend the city and they were destroyed by the Japanese in savage fighting. Older scholarship, such as Frank Dorn's The Sino-Japanese War took this as a baleful commentary on Chiang's military acumen. However, such condemnation on solely military grounds excluded the political ramifications of taking a stand and offering the Japanese fierce resistance. The KMT was on a precarious ledge in 1937 and a strong battle, even if a defeat, was vital to show the other powers like the Soviets that the Nationalists would not wither away under Japanese pressure. The throwing away of precious NRA units was a cold strategy, but it is unlikely they could have achieved different results if better deployed. The KMT's main strategic goal was to outlast Japan and it was one of the few strategies that actually was viable for Chiang. Criticism of the endemic corruption within the NRA and the overall poor performance of the KMT's military should take this strategic priority into account. The fact that the KMT resisted Japan was a victory of sorts, even if the KMT lost. Given that the KMT managed to hold out throughout the duration of the war, this strategy possesses some validity.

The CCP's guerrilla activities has also undergone a similiar transformation in recent historiography. Whereas older scholarship has taken Mao's PR campaign at face value, the general trend among contemporary historians is to be more skeptical of CCP claims. The peak of CCP activities was Battle of the Hundred Regiments in late 1940 in which CCP Eight Route Army attacked Japanese and puppet-Chinese armies in Central China. Although the CCP would use the Hundred Regiments as an example of its commitment against the Japanese, the Eight Route Army's losses prompted a drawback in massed operations, which is attested to by Vladimirov's report. Japanese counterinsurgency, the Three Alls "kill all, burn all, and destroy all," did pacify the countryside, but limited the extent to which Japanese forces could expand into Chinese territory. This counterinsurgency strategy locked the Japanese into territory it already held. From 1940 onwards, the CCP husbanded its forces and presented its limited operations as part of a larger shadow war against Japan. The general stasis of Japanese forces between 1941-44 gave Mao's declarations a degree of credibility. The Japanese Ichi-go offensive in 1944 allowed the CCP to expand into areas of northern China that the Japanese abandoned in their drive to occupy regions of southern China. This was a winning strategy for the civil war, but not exactly a mobilization of a people in arms as argued by Selden. The collapse of the Japanese military in 1945, which caught Mao by surprise, meant that the CCP was in the right place and time to utilize captured Japanese equipment and other war material. The fact that the CCP had preserved its remaining forces allowed it to present itself as a force for order in the resulting power vacuum.

So this leaves the question of who did the majority of fighting against the Japanese- the KMT or the CCP?. The problem with framing the question in these terms is that both the CCP and the KMT were not fighting a war in the sense that China's other Allies were. Of the Allies, Britain's approach most closely resembles the KMT, yet Chiang arguably receives more flak than Churchill, even when the latter pursued questionable operations to prove Britain was still in the fight, such as deploying British forces to Greece in 1941. Neither the Soviet nor American grand strategy was to hang on and outlast the Axis, but they had the resources to safely rely upon a strategy of total defeat of the enemy. Such an option was not viable for either Mao or Chiang, so they instead pursued different strategies. Both the CCP and the KMT wanted to survive the war, and they both did, rendering the question of who did the most fighting somewhat moot.

Sources

Gordon, David M. "The China-Japan War, 1931-1945." The Journal of Military History 70, no. 1 (2006): 137-182.

Peattie, Mark R., Edward J. Drea, and Hans J. Van de Ven. The Battle for China: Essays on the Military History of the Sino-Japanese War of 1937-1945. Stanford, Calif: Stanford University Press, 2011.

Spector, Ronald H. In the Ruins of Empire: The Japanese Surrender and the Battle for Postwar Asia. New York: Random House, 2007.