PDFs can do all sorts of shenanigans, such as bundle executable javascript that runs when the file is opened.
You don’t need internet to run JS though. And the JS seems to be pretty sandboxed in normal viewers too, so a pdf file shouldn’t make network requests on its own. As long as there is no iframe-like content I don’t see how rendering a document would ever need network.
if that document really does need internet access to be rendered (unlikely) then it can use the internet connection one would have used to upload the file in the first place.