Stream Manager - Activity Feed - API?

Most of the stuff that is set to the activity feed is available via EventSub or PubSub (as it’s not on EventSub yet)

So you want eventsub: EventSub | Twitch Developers

For anything missing you need a uservoice to request it.
As the activity feed is not one feed of data but a bunch of topics, so you can achieve similar with already existing topics.