You are doing it in the wrong way. An inline style attribute can only contain property declarations, you cannot declare pseudo elements inline. 
So try to write your css in a style sheet and link it to your page.
These are the pseudo elements for a textarea 
::-webkit-scrollbar              
::-webkit-scrollbar-button       
::-webkit-scrollbar-track        
::-webkit-scrollbar-track-piece  
::-webkit-scrollbar-thumb        
::-webkit-scrollbar-corner       
::-webkit-resizer
A simple example could be:
::-webkit-scrollbar {
   width: 12px;
}
::-webkit-scrollbar-track {
   -webkit-box-shadow: inset 0 0 6px rgba(0,0,0,0.3); 
   border-radius: 10px;
}
::-webkit-scrollbar-thumb {
   border-radius: 10px;
   -webkit-box-shadow: inset 0 0 6px rgba(0,0,0,0.5); 
 }
::-webkit-scrollbar {
    width: 12px;
}
 
::-webkit-scrollbar-track {
    -webkit-box-shadow: inset 0 0 6px rgba(0,0,0,0.3); 
    border-radius: 10px;
}
 
::-webkit-scrollbar-thumb {
    border-radius: 10px;
    -webkit-box-shadow: inset 0 0 6px rgba(0,0,0,0.5); 
}
<textarea id="message" rows="4" cols="50"> </textarea>