Laravel Vue.js Conditional Rendering

A lot to tackle here. First, you should set a "parent" Vue instance rather than creating a new Vue instance for individual input fields. For example, lets say you want to make the entire form a Vue instance, then when you open your form, set an id like this:

{!! Form::open(['id' => 'example']) !!}

Then, when you create your Vue instance, reference that id:

<script type="text/javascript">
    new Vue({
        el: '#example'
    })
</script>

Next, this code you have is incorrect:

{!! Form::select('mailarchive', ['-' => 'No', 'Gold' => 'Gold', 'Silver' => 'Silver', 'Bronze' => 'Bronze'], null, ['class' => 'form-control v-model="mailarchive"']) !!}

Specifically, pay attention to this part: ['class' => 'form-control v-model="mailarchive"']

What you are doing here is creating some weird class. When you specify extra HTML attributes, you need to pass an array of those attributes like this:

{!! Form::select('mailarchive', ['-' => 'No', 'Gold' => 'Gold', 'Silver' => 'Silver', 'Bronze' => 'Bronze'], null, ['class' => 'form-control', 'v-model' => 'mailarchive']) !!}

From there, another problem is how you are using v-show.

This is what you have: <div class="form-group v-show="mailarchive !='-'"">

Once again, for some reason, you are putting v-directives inside your class. Instead, use it as its own HTML attribute like this:

<div class="form-group" v-show="mailarchive !== '-'">

All that together, you should see something like this:

{!! Form::open(['id' => 'example']) !!}
    <div class="form-group">
        {!! Form::label('mailarchive', 'Mailarchive: ', ['class' => 'col-sm-3 control-label']) !!}
        <div class="col-sm-6">
            {!! Form::select('mailarchive', ['-' => 'No', 'Gold' => 'Gold', 'Silver' => 'Silver', 'Bronze' => 'Bronze'], null, ['class' => 'form-control', 'v-model' => 'mailarchive']) !!}
        </div>
    </div>
    <div class="form-group" v-show="mailarchive !== '-'">
        {!! Form::label('instance', 'Instance: ', ['class' => 'col-sm-3 control-label']) !!}
        <div class="col-sm-6">
            {!! Form::select('mailarchive', ['Select' => 'Select', '1' => 'SV01', '2' => 'SV02'], null, ['class' => 'form-control']) !!}
        </div>
    </div>
    {!! Form::submit() !!}
{!! Form::close() !!}
</div>
<script>
new Vue({
    el: '#example'
});
</script>

Here is a working example on jsfiddle: http://jsfiddle.net/zj8hwjc9/1/