How do I pass input text using v-on:change to my vue method?

First you need to define form:{email:"", ...} in the data as well.
Pass $event inside checkExist() .
Something like this,

function callMe() {
  var vm = new Vue({
    el: '#root',
    data: {
     form:{email:""},
     email:""
    },
    methods: {
       checkExist(event){

         this.email=event.target.value;

       } 
    
    }

  })
}
callMe();

<script src="https://cdn.jsdelivr.net/npm/[email protected]/dist/vue.js"></script>
<div id='root'>
 
 <input type="text" @input="checkExist($event)" class="form-control" placeholder="Email*" v-model="form.email">
<p>email:  {{email}}</p>
</div>

The v-model should already bind that input event.

But you can pass the event to v-on:change like this:

v-on:change="event => checkExist(event)"

Check exist would need to accept the event as a parameter. The value of the input will now be accessible inside the vue function via event.target.value.

checkExist: function(event){
    let value = event.target.value
}