[Vuejs]-Asynchronous pushing of values to an array in Vue.js

You actually do not need async/await at all in fetchContactPostById since there’s only one asynchronous method used, and it returns a Promise

What you need to do is return this.$http.get(.... and then remove all the async and await keywords

fetchContactPostById: function (postId) {
    return this.$http.get(this.baseUrl + '/wp-json/acf/v3/contacts-api/' + postId)
    .then(response => {
        response.data.acf.id = postId;
        let result = this.contacts.push(response.data);
        return result;
    }).catch(e => {
        this.errors.push(e);
    })
},

as an aside – the return result – do you understand it returns a number (the new length of the array) and the calling code assigns this value but never actually uses it?

the streamlined code would be

this.$http.get(this.baseUrl + '/wp-json/wp/v2/pages?slug=' + menuChildSlug).then(async response => {
      this.page = response.data[0];
      this.pageSections = this.page.acf.sections;
      this.contactSections = this.page.acf.contacts;
      for (let contact of this.contactSections) {
        await this.fetchContactPostById(contact.add_contact.ID);
      }
}).catch(e => {
    this.errors.push(e)
})

and

fetchContactPostById: function (postId) {
    return this.$http.get(this.baseUrl + '/wp-json/acf/v3/contacts-api/' + postId)
    .then(response => {
        response.data.acf.id = postId;
        this.contacts.push(response.data);
    }).catch(e => {
        this.errors.push(e);
    })
},