Styling the TwebPicker is not that easy, but there is an elegant alternative:
Set the TwebPicker visible to false, elementid to an unique value.
Add a TwebButton and style as you like
Set the OnClick of the button to:
var
filebutton: TJSHTMLElement;
begin
filebutton := document.getElementById(<elementid of the TwebPicker>) as TJSHTMLElement;
filebutton.click;
end;
Hope this is useful.