There's Neo Store[0] which is a fork of Droid-ify[1] which itself is a fork of Foxy Droid[2] (no longer updated). Personally I use Neo Store, includes more repos by default and can auto update apps.
[0]: https://github.com/NeoApplications/Neo-Store
You can get it through... regular fdroid!
From what I learned, only apps that target Android 12and above could have this background update feature. I'm on Android 13 but none of the apps I use updated. I had to switch back to using the f-droid privileged extension.
I'm on Android 13 currently, I don't have root, stock OS. All my apps update, including ones that are marked in Fdroid as "created for older Android releases". I use around 15 apps from that source and none caused me much trouble. A few times had to invoke installation manually, but honestly I think it would update automatically with more patience on my side.
That's why I'm so pumped about Fdroid Basic - it's a much better experience.
I was a user of the latter initially when it started as a fork of Foxy Droid. At one point development slowed down and Neo Store was created and had active development, and I've used that ever since. Both are still actively developed as far as I know.
My experience has been the opposite: droidify has been more "stable" than neostore for me.