For registering a subsidiary of a foreign company you would need to use an Estonian notary.

Unicount online formation service is limited by the API service that only accepts a single physical person as a founder. Estonian Business Register online service only lets you register a subsidiary of an Estonian legal person.

The list of Estonian notaries can be found here. Some of them might accept online transactions but it would still include some paperwork to prove the representation rights given by the company to you as a board member or authorized person.