Assuming you have a property (or computed, or data) which holds the information on whether the user is currently editing their profile (let's name it isEditingProfile), just provide it to input's :readonly, like so:

new Vue({
  el: '#app',
  data: () => ({
    isEditingProfile: false,
    username: ''
  })
})

<script src="https://v2.vuejs.org/js/vue.min.js"></script>
<div id="app">
  <input v-model="username" :readonly="isEditingProfile">
  
  <label>
    <input type="checkbox" v-model="isEditingProfile"> Is editing profile 
  </label>
</div>

Note: I've made isEditingProfile editable here to showcase it's working. In your app, link it to whatever logic should control it. Most probably, you need a computed for it.