I'm afraid you mix capybaras up with the coypu/ nutria.

The former has become popular in Japan and the news in Japan sometimes reports the capybara running out from their owner, but they basically live only in the zoo or as pets. The first record of their import by Japanese zoos apparently dates only back to the 1960s, though I cannot confirm this date by the academic literature for now.

On the other hand, according to the classic explanation, coypus/nutrias had been introduced in Japan to take their fur around 1930s and 1940s (roughly corresponding to WWII and its aftermath), but they often went wild after the decline of the fur industry in the late 20th century, and the Japanese Ministry of Environment designated them as an invasive alien species in 2006 (Egusa & Sakata 2009).

[Kobayashi & Oda 2016] has just challenged this classical hypothesis, and instead emphasizes the possible contributing factor of keeping coypu farming even after the WWII: The Japanese tried to consume nutria meat as a protein rich-food in the late 1940s.

If you are staying in Japan now and apparently witnessed a capybara-like animal in the wild, it will probably a coypu/ nutria.

References:

• Egusa, Sawako & Sakata Hiroshi. "Status of Coypu Control in Hyogo Prefecture." Japanese Journal of Limnology 70 (2009): 273-76. (in Japanese with a short English summary) https://doi.org/10.3739/rikusui.70.273
• Shuji Kobayashi & Sen-ichi Oda. "Coypu farming and post-WWII national policy of Japan: historical details rediscovered." Mammalian Science (Honyurui Kagaku) 56-2 (2016): 189-98. (in Japanese with English summary in the end) https://doi.org/10.11238/mammalianscience.56.189
• Japanese Ministry of Environment. "Specified Alien Species: Nutrias": also has a link to the list of related literature (in Japanese)