In the bowl of a food processor fitted with a metal blade, combine the carrots, onion, garlic, and ginger. Blend until very smooth, scraping down the sides. Add the vinegar, lime juice, and soy sauce and continue blending until the mixture is very smooth. Alternatively, you can process this mixture in a high speed blender.
Add the oils in a thin stream while the motor is running to form an emulsion. Serve immediately or refrigerate until ready to do so.