I haven't checked on the state of e-Estonia in a few months, but I think they're trying to be the place for this sort of thing? They're rolling out the services later this year I believe.
Once to file an application and then it will take up to 14 days for them to perform a background check. After which you will again have to physically appear at a police station in Estonia.
In the future this can supposedly be done through Estonian embassies but those are not exactly everywhere either.