Local Storage is not reactive, so you need to store your token in some place that is. I assume your token is being set and deleted by the application. If so, the best way to handle this is to use state like Vuex (see Vue State Management).
Option 1
If you are using local storage in order to retain the token between multiple browser sessions, the best option is to just set the token in Vuex then use Vuex persisted state to sync Vuex to local or session storage. The plugin will retrieve it again when you need to re-establish state.
Option 2
If you need it set in the local storage directly, you should include in your mutations to set/unset the localStorage when you are changing the state. This will keep your localStorage and State in sync.
For example using Vuex, if you are receiving a token in a response you can then call a Mutation to set it in Vuex state as well as set it to localStorage:
SET_TOKEN(state, payload){
state.token = payload.token
localStorage.setItem('token', payload.token)
}
You can then easily watch the Vuex state. Depending on how your Vuex is set up, it may be something like: this.$store.state.token