Heat oil in a large skillet over medium-high heat. Add pork and cook until browned, about 5 minutes, breaking it into small pieces. Add water chestnuts, green onion, soy sauce, and vinegar; cook for 5 additional minutes.

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Increase heat to high and add cabbage. Cook until cabbage is soft and liquid has evaporated, about 2 minutes. Turn off heat and stir in grated carrots. Let mixture cool for 3 minutes.

Preheat air fryer to 400 degrees F (200 degrees C).

Meanwhile, place a lumpia wrapper on a clean surface in a diamond shape, with one corner at the bottom. Place 2 tablespoons of filling in the wrapper. Fold up bottom half and then tightly fold in sides. Gently roll and seal with a couple drops of water. Repeat with remaining wrappers and filling.

Lightly mist each roll with olive oil. Arrange as many lumpia in your air fryer basket as possible without them touching. Cook in the preheated air fryer for 5 minutes. Flip and cook 3 additional minutes. Transfer to a paper towel-lined plate. Repeat with remaining lumpia.