How to add a class to vertical tabs created using Form API?
You are right vartical tabs don't have #attributes element.
But you can: 

1) place vertical tabs in #container element with class. There you can use #attributes

2) you can add #prefix, #suffix into tabs:

$tabs['#prefix'] = '&lt;div class="my-class"&gt;';

$tabs['#suffix'] = '&lt;/div&gt;';