APK of Expo app that I can put on an Android phone that doesn't have access to the Google Play Store?

If you need an APK of the Expo client app, you can download XDE and open something in the Android emulator and a copy of the latest APK will be downloaded somewhere ~/.expo in your homedir and you can use that.

Correct me if I’m wrong @jesse

Yes in ~/.expo/android-apk-cache/