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.
The good thing about mayo is that ovo vegetarians can usually make any recipe and it will turn out good.
Now, vegan mayo? That's a hard task. I'm not vegetarian, nor vegan. Tbh, I troll vegans sometimes. But, out here in real life I have a vegan cousin that is very important to me, so I've spent time learning some methods to make sure he can have meals with us when he's in town.
But I'll be damned if I've ever found a vegan mayo that didn't taste horrible, or have horrible texture. It comes down less to how well something emulsifies, and more about how there just aren't good options where you don't have a very oily result in taste and/or mouth feel. Eggs just work well for that in a way nothing else does.
I've had better luck just skipping it and not doing mayo based dishes at all. There's a ton of things you can do salads with (as in potato salad, pasta salad, etc, not just greens) that are yummy as all get out.
Usually, with vegan stuff, aquafaba gets the job done quite well once you get used to using it. But this is one thing that it fails at. None of the usual "molecular gastronomy" options have worked either.
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.