Mix soy sauce, sesame oil, ginger, and garlic in a bowl until marinade is smooth; pour into a resealable plastic bag. Add pork, coat with the marinade, squeeze out excess air, and seal the bag. Marinate in the refrigerator, 1 hour to overnight.

Heat vegetable oil in a wok or large skillet over medium heat. Add cabbage and carrot; cook and stir for 1 to 2 minutes. Push cabbage mixture aside and add pork with marinade to middle of the skillet. Cook and stir until pork is cooked through, 3 to 4 minutes. Draw cabbage into the center of the skillet; cook and stir for 1 to 2 minutes. Season with salt and pepper.