It all depends on the context. By itself, the phrase "internet movie" is using "internet" as an adjective to describe "movie." If I saw that term, I'd assume that the movie was available for streaming, but I wouldn't necessarily think, "that movie is available exclusively over the internet."
However, in a context such as the one you provide ("The actor agrees to take a role for an upcoming internet movie") I'd assume that the movie will be released as a streaming film (although that wouldn't necessarily preclude the movie from being released on DVD some time down the road).
The term doesn't infer that viewers would have to pay to watch the movie, though. If you wanted that conveyed as well, you'd have to say something like:
The actor has agreed to take a role for an upcoming pay-per-view internet movie.
The actor has agreed to take a role for an upcoming internet movie, which will be available to subscribers only.
It's a complicated question, because word choice depends on the available payment options, too.