I may be missing something so apologies if I'm telling you something you already know but: there are plenty of web search APIs for agents - the Brave one is the one I use: https://brave.com/search/api/
Tavily one is free, too. Easy to use cli with a skill.
Most web search APIs offer a MCP with limited free usage, but for production use you still need to pay.
I already know the Brave Search API, but thanks for telling me. The issue is that it's expensive. It's 5 USD per 1000 requests, while DeepSeek's is 0.75 USD per 1000 requests.
I gave it a kick and got mixed results. For example, searching for a Microsoft product couldn't find its landing page, and searching for a niche open source package landed in the Debian repos for the top hits, but the home page was in the first 10 hits.
Lovely to see they're doing their own search though, I love how many folk are actively working on search clones at the moment.
I haven't really tested it. But they might be doing keyword search like ceramic.ai instead of semantic search, so that's why the results might be worse, but I'm not sure.