As Sony promised it would last week, it has released the PlayStation Vue app for Android today, featuring real-live cast support. The app is available on the Play Store now, or via APK Mirror right here. To use Vue, you'll have to be an existing Vue subscriber - the Android app doesn't have any kind of sign-up flow, so if you want to start a Vue trial, you'll need to do that through your PlayStation 3 or 4 console, or through the PS Vue webpage. You do not technically need a PlayStation to use Vue, and Sony offers a free 7-day trial, but you will need a supported Vue TV device (Amazon Fire TV and Roku, for example) to set up your initial profile - the Vue Mobile app does not support this feature.
The app also appears to be installable on Android TV devices, but it doesn't actually work - the app installs, but you can't open it. Probably just an oversight on Sony's part, the app probably wasn't meant to be available for Android TV. Though if Vue does actually come to Android TV, we'll let you know.