Antbot.pw provides a free, open API endpoint for checking a domain or email address against a frequently-updated list of disposable domains. CORS is enabled for all originating domains, so you can call the API directly from your client-side code.
GET https://antibot.pw/api/[email protected] HTTP/1.1
The response will be JSON with one boolean property, e.g. {"disposable":false}
Using jQuery?

<script>
    $( "#email" ).change(function() {
      var val = $("#email").val();
      $.get('https://antibot.pw/api/disposable?email='+val,
        function (data, textStatus) {  
            if(data['disposable'] == true){
              alert("email disposable");
            }
      });
    });
</script>