I ran a YaCy instance for a while like a decade ago. It does federate index requests, and when you search it propagates the search request across a bunch of nodes. When my node came online it almost immediately started crawling stuff and it did get a bunch of search queries. But the network was still pretty small back then and the search results were… not great. That’s the price of independence from Google’s and Microsoft’s giant server farms, it’s hard to compete with that size.
But at the rate Google and Bing are enshittifying, I think it’s worth revisiting.
Using ActivityPub for this would be immensely wasteful. It’s just not feasable that all instances would have the whole index because it’s so large. Back when I tried it, the network still had several TBs worth of indexed pages. This is firmly in the realm of distributed P2P systems. One could have an ActivityPub plugin however to receive updates from social media near instantly and index those immediately with less overhead. But you still want to index wikipedia, forums, blogs, whatever the crawlers can find.