I guess one of the largest collections of Estonian movies and TV series is available on the websites of Estonian national TV channels ETV
There are these sections called saadete järelvaatamine
(afterviewing of shows):
Unfortunately, the English versions of these pages seem to be non-existent, and I am not sure at all how much of it (if any) is viewable outside of Estonia, so let me know if you try it out
I guess some shows there have also somekind of time limits of how long are they up there, but I haven't really dug into it
It should be possible to also watch something on the websites of some commercial channels, like Kanal 2
, but I haven't tried it myself yet.