This depends to some degree on what you call an invasion. For example, was Germany's taking of Czechoslovakia an invasion? That said, each of the three Axis countries could be said to have invaded at least one other country.
Germany can be said to have invaded Czechoslovakia. There was essentially no fighting involved, though, because the Czechs had surrendered before Germany troops entered the country. Germany can also be said to have invaded Austria in the Anschluss, though that was fairly consensual.
Italy invaded Ethiopia. It did this in 1935. This was the only country Italy invaded. Italy also held other African countries like Libya, but took them well before WWII.
Japan invaded China. It was at war with China for years before WWII started. As part of this, it created its own puppet state of Manchukuo in Manchuria.