I will admit up front that this answer does not answer the original question, that of generally being able to access the Manifest. However if what is really required is to read one of a number of "standard" Manifest attributes, the following solution is much simpler than those posted above. Note that this solution is in Kotlin, not Java, but I would expect that a port to Java would be trivial. (Although I admit I don't know the Java equivalent of ".`package`".
In my case I wanted to read the attribute "Implementation-Version" so I started with the solutions given above to obtain the stream and then read it to obtain the value. While this solution worked, a coworker reviewing my code showed me an easier way to do what I wanted. Note that this solution is in Kotlin, not Java.
val myPackage = MyApplication::class.java.`package`
val implementationVersion = myPackage.implementationVersion
Once again note that this does not answer the original question, in particular "Export-package" does not seem to be one of the supported attributes. That said, there is a myPackage.name that returns a value. Perhaps someone who understands this more than I can comment on whether that returns the value the original poster is requesting.