Are the Self-Defense Forces a military organization?

No, The Constitution of Japan declares permanent renouncement of war in Article 9 of the Second Chapter.
The postwar interpretation of this article was that Japan would never start a war or resort to military force to solve an international dispute, but that the country had not given up the right to defend itself.
This was the main reason why the Self-Defense Forces originated in 1950 as the National Police Reserve.
This was reorganized as the National Safety Forces in 1952, developing in 1954 into the Self-Defense Forces with the enforcement of the Self-Defense Force Law.
Although the SDF, which consists of ground, maritime, and air self-defense forces, has fighter planes, tanks and surface-to-air missiles, it is not actually referred to as a military organization.
As a result, whenever an overseas dispatch is made to provide humanitarian and recovery assistance, there is much discussion.
Because the Self-Defense Forces are not an "army" they are not allowed to go to combat zones.
Although the Diet decided to send the Self-Defense Forces to help in the recovery effort after the 2003 Iraq War, many people opposed sending forces to an area of terrorism and resistance, making it impossible to separate the battle zones from the battle-free zones, despite the war being declared over.