When I ran a warehouse full of people, each one had an A4 card with 28 squares on it, they then filled in the date or dates they wanted off, I signed them to authorise then put them onto my year planner.
Their cards were kept in a filing cabinet we all had access to, that way they could easily see how many days they had used/booked/left and no arguments were needed.
HR also had a form they needed filled out and they then wanted to approve the holidays for my staff!! That didn't go down well with any of us, I used to put the form through once they had started their time off.