I edited your fiddle but as I never moved in it I do not know if the change I made will appear to you.
here the link
What would you do to solve your problem:
I would put the messages in a second internal div, gave the message id of them:
<div id="chat">
<div id="mensagens">
chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>chat
<br>
</div>
</div>
and in the css would set the distance with a margin, css stayed like
#chat {
height: 39vh;
width: 27.5vw;
display: block;
overflow: auto;
position: fixed;
top: 55vh;
left: 21vw;
background-image: url("http://pre11.deviantart.net/b65d/th/pre/i/2005/076/5/e/frame_by_anilestock.jpg");
background-size: 100% 100%;
-margin:auto;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
}
#mensagens
{
margin-left:30px;
margin-top:20px;
width:120px;
height:139px;
overflow:auto;
}