Daniel Farrell's Bootstrap Combobox does the job perfectly. Here's an example from his GitHub repository.
$(document).ready(function(){_x000D_
$('.combobox').combobox();_x000D_
_x000D_
// bonus: add a placeholder_x000D_
$('.combobox').attr('placeholder', 'For example, start typing "Pennsylvania"');_x000D_
});
_x000D_
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.7/css/bootstrap.min.css">_x000D_
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/bootstrap-combobox/1.1.8/css/bootstrap-combobox.min.css">_x000D_
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.2.1/jquery.min.js"></script>_x000D_
<script src="https://cdnjs.cloudflare.com/ajax/libs/bootstrap-combobox/1.1.8/js/bootstrap-combobox.min.js"></script>_x000D_
_x000D_
<select class="combobox form-control">_x000D_
<option></option>_x000D_
<option value="PA">Pennsylvania</option>_x000D_
<option value="CT">Connecticut</option>_x000D_
<option value="NY">New York</option>_x000D_
<option value="MD">Maryland</option>_x000D_
<option value="VA">Virginia</option>_x000D_
</select>
_x000D_
As an added bonus, I've included a placeholder in script since applying it to the markup does not hold.