Home.vue

<script>
import Vue from 'vue'
import axios from 'axios'
import VueAxios from 'vue-axios'
const qs = require('qs')

Vue.use(VueAxios, axios)

export default {
  name: 'home',
  data () {
    return {
      token: '',
      errors: [],
      data: [],
      id: [],
      solicitacao: [],
      status: []
    }
  },
  methods : {
    getToken() {
      const apiAuth = `url`
      
      axios.post(apiAuth, qs.stringify({
          client_id: '' ,
          client_secret: '',
          email: '',
          password: ''
        })).then(response => {
          this.token = 'Bearer ' + response.data.token
        }).catch(e => {
          this.errors.push(e)
        })
    },
    getOutraCoisa() {
      const apiReq = `url`
      
      axios.post(apiReq, {}, { headers: { Authorization: this.token } })
        .then(response => {
          console.log(response.data)
        })
        .catch(e => {
          this.errors.push(e)
          console.log(e, this.token) // TOKEN CHEGA AQUI
        })
    }
  },
  mounted() {
    this.getToken()
  },
  watch : {
    // Quando o valor de this.token mudar, vai disparar a nova request
    token : function() {
      this.getOutraCoisa()
    }
  }
}
</script>

Valeu pela força, Adalberto. 😀