If you hook up on the elements scroll event, you could do something like this, where you set the inner to width: 100% and a padding-right: 20px. (the padding right value need to be bigger than the scrollbar is wide) 
That will push the scrollbar out of view, and on scroll you remove the padding, init a timer which will reset the padding if one stop scrolling
(function(timer) {
  window.addEventListener('load', function() {
    var el = document.querySelector('.inner');
    el.addEventListener('scroll', function(e) {
    (function(el){
      el.style.padding = '0';
      clearTimeout(timer);
      timer = setTimeout(function() {
        el.style.paddingRight = '20px';      
      }, 100);    
    })(el);
    })
  })
})();
.outer {
  height: 180px;
  width: 500px;
  border: 1px solid green;
  overflow: hidden;
}
.inner {
  width: 100%;
  height: 99%;
  overflow: auto;
  padding-right: 20px;
}
<div class="outer">
  <div class="inner">
    Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some
    content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some
    content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>
  </div>
</div>
 
 
Updated, toggling a class, using Element.classList, instead of an inline style, which might be a more recommended way.
(function(timer) {
  window.addEventListener('load', function() {
    var el = document.querySelector('.inner');
    el.addEventListener('scroll', function(e) {
    (function(el){
      el.classList.add('scroll');
      clearTimeout(timer);
      timer = setTimeout(function() {
        el.classList.remove('scroll');
      }, 100);    
    })(el);
    })
  })
})();
.outer {
  height: 180px;
  width: 500px;
  border: 1px solid green;
  overflow: hidden;
}
.inner {
  width: 100%;
  height: 99%;
  overflow: auto;
  padding-right: 20px;
}
.inner.scroll {
  padding-right: 0;
}
<div class="outer">
  <div class="inner">
    Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some
    content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some
    content<br>Some content<br> Some content<br>Some content<br>Some content<br> Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>Some content<br>
  </div>
</div>