The Wikipedia entry for kamikaze says flatly that it was not the Japanese term:
The Japanese themselves did not use the word Kamikaze to refer to these World War II attacks. The official Japanese term was tokubetsu kōgeki tai (特別攻撃隊 “Special Attack Units”). The word Shinpū (also meaning “divine wind”; just another reading of the same kanji for kamikaze) was also used informally for suicide units. U.S. translators erroneously used the Japanese word Kamikaze, which has a similar original meaning of “divine wind” (see Kamikaze typhoon).
Later it explains that “The word kamikaze originated as the name of major typhoons in 1274 and 1281, which dispersed Mongolian invasion fleets,” which I had known. But the business about the two readings (compare hara-kiri/seppuku) intrigued me. Unfortunately, when I investigated further, things got murky; this site says
The two Japanese characters (kanji) for “kamikaze” (meaning “divine wind”) can be read in two ways: “kamikaze” or “shinpu.” Nagatsuka speculates that nisei (second-generation Japanese-Americans) in the U.S. military were the first to use the pronunciation “kamikaze” to describe the special attack suicide squads because “they did not know how to read Japanese correctly and so pronounced the two Japanese characters for Divine Wind in a more vernacular way [kamikaze]” (p. 142). He cites no support for such an assertion. Although Shinpu was the official name given to the first unit formed in the Philippines in October 1944, people in Japan both during and after the war frequently read the two kanji as “kamikaze.”
I added a  tag, but being too impatient to wait for some Wikipedian to notice and respond, I thought I’d ask you all: anybody know whether ordinary Japanese used the term kamikaze during the war or whether it was imported from ignorant Yanks afterwards?