Codex run locally doesn't have access to any "web search" tool, but that doesn't stop it from trying to browse the web via cURL from time to time. Place hyperlinks in the documents in the repository and it'll try to reach them best it can. It doesn't seem overly eager doing so though, and only does that when absolutely needed and it can't find the information elsewhere. This has been my experience with gpt-5-high at least.
Probably a security feature. If it can access the internet, it can send your private data to the internet. Of course, if you allow it to run arbitrary commands it can do the same.
