External modules like axios would need to be bundled into your chrome extension in order for it to work as you intend. You can use a bundler like webpack in order to do this, but there is a decent amount of configuration needed to get up and running.

I would recommend using the fetch API as an alternative. It's already freely available in chrome extensions and works in very similar way to axios.

A fuller code example comparing fetch to axios:
https://blog.logrocket.com/axios-or-fetch-api/