Place the eggs in a large pot and cover with water. Bring to a boil, then remove from heat, cover, and let sit for 12 minutes.
Drain the eggs and transfer to a bowl of ice water. Once cool, peel and slice each egg in half lengthwise.
Scoop the yolks into a mixing bowl. Mash them with a fork, then stir in the mayonnaise, Dijon mustard, white vinegar, salt, and black pepper until smooth.
Spoon or pipe the yolk mixture back into the egg white halves.
Dip a toothpick into the soy sauce and draw three or four concentric circles on top of each filled egg. Drag the toothpick through the circles at even intervals to create a web pattern.
Slice each black olive in half to make the spider bodies. Cut the remaining halves into thin strips for the legs. Place one olive half in the center of each web and arrange the strips around it as legs.