How do you find a developer of an app that hasn't been updated?

Open your App Store purchase list and find the app on that list. From there you can find the developer information and try to download it again (or see why it disappeared).

Look in the AppStore app.

At the top right there is an icon representing yourself - which you can set so you might recognise it easily.

Press that to go to your account then choose Purchased and then My Purchases to see all the apps you have bought

Search for your app and you will get to the App's page.

That should have details of the developer etc.

But I expect they have stopped responding. However their might be enough information to search the web for their current whereabouts.