No, they don’t. St Lucia citizens may visit any of the Schengen member countries, which includes 22 EU countries along with Iceland, Liechtenstein, Norway, and Switzerland. They are allowed to stay there for 90 days out of 180 days.
St Lucians need a valid passport to visit the Schengen Area. This passport must be valid for at least three months after they exit the Schengen Area.