I think you need to understand how React Native works in order to understand.
What RN actually does is to expose native API to your Javascript code base. Basically a RN App is composed by 2 things:

As you can immagine the Native part is everything that is written in Swift/ Objective-C/ Java
and that is the part you can not update without going through the App stores.
Now the interesting thing is the JS part, remember we said that basically you are consuming native API with JS. If you notice when you run react-native run-ios
or react-native run-android
a server is instanced which serves a bundle to your emulator / physical device.
Now if you think about it basically when you open the App the bundle is downloaded an then run. When you update your codebase while the app is running on the emulator the servers sends a signal trough the socket to notify the client that an update is available. At that point the client downloads the bundle and the app is reloaded.
Now to answer your question, yes you can serve the JS Bundle on your server and make the app check when is loaded or resumed if a new version of the JS Bundle is available, if so to download it (OTA update). As you can understand only the Javascript part can be update in this way and not the native part as well.
Then again, there are a few services that already do this like codepush by MS.