In a small bowl, place the sun-dried tomatoes, cover with them with the hot water, and set aside to rehydrate for 10 minutes. Transfer the rehydrated sun-dried tomatoes and their soaking liquid to a blender or food processor. Add the basil leaves and garlic, and pulse a few times to roughly chop everything. Add the soy mayonnaise and balsamic vinegar, and process to well combine.
Transfer aioli to an airtight container and store in the refrigerator for up to a week.