The Council of the European Union has updated its recommendation on travel from third countries, and removed Japan from Safe Country’s List, meaning non-essential travel (tourism) is no longer possible. See the current border restrictions and exemptions here: https://vm.ee/en/information-countries-and-self-isolation-requirements-passengers/
If for some reason entering the country is necessary, the Embassy recommends to contact PPA directly (Police &Border Guard Board) and apply for a special entry permit.
The application must be sent to firstname.lastname@example.org. The purpose of the special application is to allow third-country nationals to apply for crossing the external borders, for the purpose of entering Estonia under special circumstances.
The application for permission to cross the border in exceptional circumstances and more information can be found on the website of the Police and Border Guard Board: https://www.politsei.ee/en/instructions/emergency-situation