Ingredients:
1 egg
1/2 teaspoon salt
1 tablespoon acid (I used lemon juice, but vinegar works too)
1+ chipotle in adobo sauce (I used 4)
3/4c olive oil
Directions: Put all ingredients in a wide mouth jar or beaker if you have one, ensuring your opening is large enough for your stick/immersion blender
Fully submerge a stick blender and begin blending starting at the bottom, after a short time start slowly moving the blender upward until everything is blended and the oil has emulsified into mayo.
This likely could be made vegan by replacing the egg with another emulsifier, I think lecithin will work. I am not familiar with using it, so use at your own risk.
Mayo uses an egg as the emulsifier, thus not vegan. Swap the egg for garlic and you have aioli which can be vegan. There are other emulsifiers you can use too.
Also, stores have decided to ignore these differences and most use aioli to just mean garlic mayo but that's it's own problem.