How do you mean 'play a part in WWII?' In what sense?

I would say that the biggest influence ancestral/ethnic heritage played in WWII was in the case of the 442nd Regiment - sometimes referred to as the Japanese American regiment, or the Nisei Regiment.

The attack at Pearl Harbour led to the internment of Japanese-Americans all over the West Coast of America, in areas deemed important to military security. In Hawaii however, Americans of Japanese background were vital to the economy, quite simply because there were so many of them there. Removing them would create a vacuum. Japanese-Americans were mistrusted in ways that German and Italian Americans never were. They initially weren't allowed to serve in the American forces, despite the fact that many of them were Nisei, or children of immigrants. In other words, these were people with Japanese ancestry who had never seen Japan, were old enough to fight(so had 2+ decades in America), but weren't trusted.

In 1943, the ruling was reversed, and the 442nd was formed with volunteers from Hawaii and the West Coast. The resultant regiment was something like 99% Japanese American, and went on to become one of the most decorated military units in American history.

However, the 442nd never served in the Pacific theatre! They were only deployed in Europe, where there was no chance of running into the Imperial Japanese forces. On the other hand, German and Italian Americans served wherever possible, with no fear that they would suddenly turn on the Americans.

Miscellaneous:

Daniel Inouye, one of the longest serving senators in American history, and an undefeated politician for half a century served in the 442nd.

Dr Seuss drew racist comics that played on fears of Japanese espionage by the Japanese-American community.

Do you mean play a part in colloqueal interaction during the time of WWII? I can't see it play a large role in the war itself.

Give a holler if you want me to type out a detailed answer about the shift from race to ethnicity as a focus on the American homefront and the a curious multi-ethnic American ideal that was wildly out of step with the realities of segregated midcentury America. (Essentially black people get erased from narratives of what it means to be American, while propaganda films celebrate that there could be an Italian, and Irishman, a Southern boy, and a New Yorker all on one aircrew or production line together.) But I basically only do homefront, sorry.