Looks like you need an id card, and it looks like this is something issued to Estonians. If the question is whether anyone (not just Estonians) can easily and quickly setup a company there, the answer looks to be no.
You can become an "e-Resident" and receive an ID card. I believe this requires you to travel to Estonia once to get the card, however they are working on rolling this out to their embassies in other countries.