I am having an issue with trying to connect the WP Rest API with my local install of WordPress. When I run npm start and then type https://localhost:9000/ into Chrome and FF (with my local WP install running via Flywheel) I am getting a warning that says that the connection is not private/secure. My local WP install url is http://apidemo.dev/wp-json and I am assuming that this (..being http and not https) may be the issue.
In local you can enable SSL. It will still show a warning saying is not secure bc the certificate is self signed but you should have an option to proceed anyways.
On another note I had issues with Local in NGINX mode not displaying the /wp-json/ properly but asking in the local support forums resolved this issue (I think it was switching off NGINX mode)