@Jared -- Allowing file downloads from an embedded app is not compatible with the security policies of ArcGIS StoryMaps (nor is it compatible with the policies of many other web pages where a dashboard or other app that allows downloads might be embedded). Although we understand the workflow improvement this could provide, this is not something we are looking to allow at this time.
If there is an app that provides a download feature that does not work while embedded, our current recommendation is to provide a link so that the application can be opened and used in a new browser tab so the download will work as expected.
Can you clarify which app you are embedding? There are ways for apps to do secure downloads while embedded via pop-up windows. We can try to talk to the folks on that team to see if they are able to provide this as an enhancement. It would also help if you added an idea to that app's ideas board with this request too, so others can vote on it.
Owen Evans
Lead Product Engineer | StoryMaps