Does any vector database like Qdrant exist for the browser/IndexDB?

none, as far as I am aware

Why it should be in the browser and not just make API calls from js?

Are you looking for something like VDBLite?

https://community.openai.com/t/its-like-sqlite-but-for-embeddings-introducing-vdblite/19204

Trying to build a client-side chrome extension that has access to plenty of sensitive information I don't want to be responsible for.

Along these lines, but something accessible via JavaScript would be ideal—something in browser.

I'm not aware about your use case, you can try "native messaging"?