What is gapps, some open source component, library, app? No? Then no, it’s not needed.
Jokes aside…
If you are on Android 12 or later, and not on a Xiaomi device, then autoupdates should just work.
Be sure to slide both Wi-Fi and mobile data all the way to the right.
Be sure to turn both “auto” toggles in Settings.
Also do update to 1.20 just to be sure, expand Versions… 
/LE: if you read this in the future, note that there’s no restriction anymore for any device, so if on Android 12 or later it should work, update to the latest F-Droid Client or Basic