As far as I know a Jew is allowed to buy a black unflavored coffee in a disposable cup anywhere, and there is no kashrus problem.
As far as the milk goes, if you drink chalav stam milk, you would be allowed to add that to your coffee.
If you do not eat cholov stam, Starbucks has pareve soy milk with a hechsher on the container. I have asked to see the container in many different Starbucks around America (NY, California, Ohio, Maryland…), and they all seem to use the same brand that has the same hechsher. I would still ask to see the container each time though, just in case.