The east empire company isn't actually part of the empire, they are a company that does trade PRIMARILY in imperial ports, but they also go where the profit is at, and even stormcloaks agree there business helps ALL of skyrim.
Pretty much this. The Stormcloaks would be pretty stupid to cut all
economic ties with the Empire and the other provinces. That would tank their economy and probably ultimately set off another rebellion since starving, impoverished people are generally unhappy people, and see removing the rulers who got them into that situation in the first place as a good solution. See also: what happened to Imperial Russia.
Something tells me the Stormcloaks aren't so dumb as to seize power and then immediately commit economic suicide. The EEC may face a number of restrictions and sanctions, but they'll still be allowed to trade in and with Skyrim.
...For pete's sake, they have a warehouse in Windhelm. I don't think the Stormcloaks are going to toss them out anytime soon.
This. How can you deny the EEC is not affiliated with the empire when it even has the word "empire" in the name. For all we know maybe the empire may use it as some sort of spy network to help retake skyrim? Though that plan svcks.
For the same reason the real life East India Company took the name they did. They were formed to trade primarily in the
East Indies, hence the name. (They actually wound up trading mostly in India. Name still works though.)
The EEC, when it started, presumably traded in Imperial ports and provinces (I imagine that's changed somewhat, with the last 200 years or so being one big secession party for a lot of the former Imperial provinces), which is why they picked the name they did. They may well have--or had, during the Septim Empire--some sort of imperial patronage. That's definitely not unheard of in RL history, either. And it would explain how the Emperor's cousin has a top position in the company. Nepotism lives indeed.
However, that doesn't mean they work solely at the service of the Medes, or that they're all Imperial spies. They're still an independent organization, despite the misleading name. They're capitalists first and foremost, and imperalists second at best. I'm sure there are a few Imperial spies within their ranks, but certainly not enough to stage a takeover of a Stormcloak-led Skyrim.