Place the eggs in a saucepan and cover with cold water. Bring to a boil over medium heat, then remove from heat, cover, and let sit for 12 minutes.
Drain the eggs and transfer them to a bowl of ice water. Cool for 5 minutes, then peel under running water.
Slice each egg in half lengthwise and remove the yolks to a mixing bowl. Arrange the whites on a serving plate.
Mash the yolks with a fork. Add mayonnaise, mustard, vinegar, salt, and black pepper. Stir until smooth.
Spoon or pipe the yolk mixture back into the egg whites.
Slice 6 olives in half lengthwise to create spider bodies. Cut the remaining olive pieces into thin strips for legs. Place one olive half on each deviled egg and arrange the strips as legs.
Sprinkle paprika lightly over the filling.