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/
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.
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.