For making authenticated API requests from a third party app, you’ll need to install a plugin to give you different methods of authentication.
The most convenient but less secure is Basic Authentication: https://github.com/WP-API/Basic-Auth, it’s appropriate for a local development environment.
This allows you to make authenticated requests by passing username and password in the body of the POST request.
I don’t have an example in Vue, as I’m not familiar with its syntax.