Does the East Empire Trading Company still ship to Raven Rock?

No. They abandoned Raven Rock when the ebony mine dried up.

Though they still have trading operations in Blacklight, Necrom and Tear.

With the mine reopened, they could reintegrate Raven Rock into their shipping routes, since the Dunmer probably wouldn't be able to organize the ebony trade all by themselves.

But in the game's timeframe, no, the Company isn't involved in Raven Rock at all.

Why not?

I think there's one NPC commenting on how the reopening of the mines will bring the ebony traders back to Solstheim. I think the Dunmer are cool with the idea of having the East Empire company back, should it happen. They're not famous merchants or traders and, besides, I think they own a debt of gratitude when the empire gave that land away to the refugees.

House Dres disagrees with you

Even while they do not specifically mention the Company, I think it is indeed safe to assume that only the company has the necessary resources to trade it.

But it's not the Empire who gave them them Solstheim, it was the High King of Skyrim. The modern-era Dunmer should instead hate the Empire. First they occupied Morrowind, then, when the Dunmer disbanded most of their own forces due to the Legions taking over, the Imperials abandoned the province when the Dunmer needed them most. First during the Oblivion Crisis, then the Red Year, then the lizard Invasion.

Currently the Dunmer need any money they could get for rebuilding their homeland and the Empire is in dire need of weapons to rebuild its legions. It's just business, really.

House Dres doesn't exist any longer...

Oh how wrong you are

http://www.uesp.net/wiki/Lore:House_Dres

It;s around we just don't know it's condition.

Also Morrowind was independent after the fall of the third empire, the empire had no responsibility to help them or to aid them against the argonians.