While you won’t be able to use multiple orgs at once, you could look into “switching” orgs by doing silent auth and including the organization param. You need to make another /authorize request with the new org_id in order to specify the “new” org. While I haven’t attempted this using the vue SDK myself, the following topic may be useful:
RT but this doesn’t seem to be easily done with auth0-spa-js and Vue, since it is attached to the Vue instance I think I would have to setup a whole new Vue instance.