Luckily our friends at Expo have a great solution for us. If you build your entry using React Native, and publish it to Expo, you'll only need to post your "expo.io" project link as your entry. Judges just need to install one app, and then Expo provides a landing page where any app can be loaded within in (using a qr code or by entering the url).
If you’re new to React Native entirely, the easiest way to get ramped up on React Native is to follow guides at http://reactnativeexpress.com/ before jumping in to Expo.
When submitting your application for judging, just use your project's published Expo url, e.g. "https://expo.io/@community/native-component-list"
This is what your project install page will look like:
If you decide to build a React Native app this weekend, please make sure to also upload a video demo of your app, and a screenshot (our automatic screenshotting service won't really work, it'll just snag a screen of the Expo page).