After the great work bringing video recording into the Camera component, it would appear that the only thing left to complete the "visual media" suite would be to allow a user to select videos from their phone's storage. This should work just like the current Expo ImagePicker but filter to only video files (and/or both, maybe).
Unfortunately I don't know about Android, but on iOS the RN ImagePicker (https://facebook.github.io/react-native/docs/imagepickerios.html) allows a developer to specify the type of media (images and/or videos), whereas the Expo ImagePicker has it hardcoded to images only (https://github.com/expo/expo/blob/master/ios/Exponent/Versioned/Modules/Api/EXImagePicker.m#L98).
Using the native iOS ImagePicker in RN would be fine in Expo but the FileSystem API doesn't support the URIs returned from it (they are not inside the documents or the cache directory for the app, plus Expo has extra validation on paths to namespace the Expo app too), so we can't access the files without ejecting and getting some native FS access. While an update to the FileSystem API would help (as per @Brent Vatne's suggestion to me in the Camera component request), I really feel that because we have ImagePicker in Expo, the video picker should also be part of Expo, including the magic Expo does to make these files natively accessible in the cache directory after they are selected (https://github.com/expo/expo/blob/master/ios/Exponent/Versioned/Modules/Api/EXImagePicker.m#L138).