How to inherit Bearer token from Vue app deployed on App Engine?

I have a Vue web app deployed on App Engine. Because its deployed on App Engine, authentication is already provided and requires no configuration on my end.

The web client needs to send a OAuth 2.0 Bearer token along in a request when uploading to a Google Cloud Storage Bucket like this:

<script>'', file,
          headers: {
          'Authorization': 'Bearer TOKEN'
          'Content-Type': 'application/json'
        }).then((response) => {
          console.log('response is: ', response);
        }).catch((err) => {
          console.log('error is: ', err);

I've looked at the documentation from the google-api-javascript-client, but I am not starting an auth flow from scratch.

I've also added the App Engine Service Account as a member of the bucket and provided it with permission to create objects in the bucket

Since the user will need to authenticate with their Google credentials in order to access the home page, how can inherit this Bearer token to send an authorized request to upload the item to a bucket?