The combination of a form builder that outputs html and changing the values with vuejs based on other form fields is not going to work well. It is possible with just plain javascript or jQuery. But VueJS won't do anything for you here if you use the html form builder.
You can create your own component that creates this form for you. You can pass the required data for the selects as propsand than render your form with vuejs. Than you can easily populate the select fields based on another input field.
This requires some vuejs experience. There must be some examples out there that should give you a good start but I can't find one at this moment.
update see for a simple example https://codepen.io/anon/pen/aJOOOe?editors=1010 found in SO ticket: https://stackoverflow.com/questions/42470501/vue-js-dependent-select