Here's a question on how to center a <select> scroll bar to the selected item.

I'd like to find a way to do the same thing for a Bootstrap Dropdown Menu so that the menu automatically scrolls to the .active list item.

Here's some code:

<ul>
  <li class="dropdown switch">
    <a href="active" class="dropdown-toggle" data-toggle="dropdown" id="switch-a">
      <span>Name</span>
    </a>
    <ul class="dropdown-menu switch-items" id="switch-dd">
      <li>...</li>
      <li>...</li>
      <li class="active active-dd"></li>
      .
      .
      .
      <li>...</li>
    </ul>
  </li>
</ul>

Is this possible or do I have to a HTML <select> and spending time styling it?

Not only is it possible, it's actually much easier to deal with native HTML objects than it is to deal with the mysteries and inconsistencies of styling the <select> element.

You just have to listen to the dropdown open event with shown.bs.dropdown.
Once it's open, just find the only .active item and call .focus() to bring it into view.

Like This:

<!-- language: lang-js -->
$(".dropdown").on("shown.bs.dropdown", function() {
   $(this).find(".dropdown-menu li.active a").focus()
});

Demo in Stack Snippets:

<!-- begin snippet: js hide: true --> <!-- language: lang-js -->
$(".dropdown.focus-active").on("shown.bs.dropdown", function() {
   $(this).find(".dropdown-menu li.active a").focus()
});
<!-- language: lang-css -->
.scrollable-menu {
    height: auto;
    max-height: 200px;
    overflow-x: hidden;
}
<!-- language: lang-html -->
<link href="//cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.2/css/bootstrap.css" rel="stylesheet"/>
<script src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.3/jquery.js"></script>
<script src="//cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.2/js/bootstrap.js"></script>

<div class="container" >

  <div class="dropdown focus-active">
    <a href="#" class="btn btn-primary"
       data-target="#"  data-toggle="dropdown" 
       aria-haspopup="true" role="button" aria-expanded="false">
      Dropdown trigger
      <span class="caret"></span>
    </a>

    <ul class="dropdown-menu scrollable-menu" role="menu" aria-labelledby="dLabel">
      <li><a href="#">Item 1</a></li>
      <li><a href="#">Item 2</a></li>
      <li><a href="#">Item 3</a></li>
      <li><a href="#">Item 4</a></li>
      <li><a href="#">Item 5</a></li>
      <li><a href="#">Item 6</a></li>
      <li><a href="#">Item 7</a></li>
      <li><a href="#">Item 8</a></li>
      <li class="active"><a href="#">Item 9</a></li>
      <li><a href="#">Item 10</a></li>
      <li><a href="#">Item 11</a></li>
      <li><a href="#">Item 12</a></li>      
    </ul>
  </div>

</div>
<!-- end snippet -->