The EU-wide rules apply in Estonia the same way they apply everywhere else — those are on the rules page. This page carries its producer register and the authority responsible for packaging there.
Where businesses register
Estonia keeps its list of packaging businesses in Pakendiregister, the packaging register. The rules call this a producer register. Being on it is what allows packaging to be made available in Estonia at all, so it is usually the first thing a business shipping there deals with.