I don't believe you can cause the scrollbar to overlay the site, but you can force it to always be visible by setting (demo):
body{
overflow-y:scroll;
}
This is on the assumption it is your body element that requires scrolling, alternatively simply set it on the element representing the scrollable viewport.
The overflow-y CSS property specifies whether to clip content, render a scroll bar, or display overflow content of a block-level element, when it overflows at the top and bottom edges.
scroll The content is clipped and desktop browsers use scrollbars, whether or not any content is clipped. This avoids any problem with scrollbars appearing and disappearing in a dynamic environment. Printers may print overflowing content.