Not the whole web; LinkedIn and a few others block us and we fully respect robots.txt, but we have ~8 billion pages.

edit: from article, "Doing this for a few urls is easy but doing it for billions of urls starts to get tricky and expensive (although not completely out of reach)" - indeed so, but we have now done embeddings for about half of those ~8 billion pages and are using them for mojeek.com.

We have an API with many features including uniquely authority and ranking scorings. Embeddings could be added.

https://www.mojeek.com/services/search/web-search-api/ used by Kagi, Meta and others. Self-disclosure; Mojeek team member.

Brave Search also has one https://brave.com/search/api/

Perplexity has models that include RAG access available via API - their "online" models. https://docs.perplexity.ai/docs/model-cards

Kagis API is very limited relative to what features they offer through their web interface. You can't search using the full results they serve. They only offer their index as an API, which I think is relatively small.