You have found an interesting ambiguity.
Solely is an adverb modifying the predicate adjective responsible. Omitting solely we have
NPR is responsible for its content.
As you imply, there are at least two interpretations for solely.
NPR alone [and no one else] is responsible for its content.
and
NPR is responsible only for its content [but not for its style/financing/political fallout, etc.]
and possibly,
NPR is responsible only for its content [but not for the content of others]
I think most listeners would intuit the first (and I think that is what is intended), but the latter two are not logically or grammatically excluded.