Well... This company has not been created yet. The Empire of Reman didn't manage to conquer Morrowind. They were close, but... No. So, the Empire didn't put its hand on Ebony mines and Dwemer artifacts and so on. Yet we have an "ancestor" of the EETC on Gold Coast. No spoilers - but some quests and lore books are revealing its activities. Also, check Seyda Neen NPC dialogues
This the earliest I can find for EETC.....
When exactly the East Empire Company was founded is unknown, but it first appeared in Morrowind. It is however known that it has existed at least for 5 centuries in 4E 201, meaning that it has existed since at least 3E 134.