All of the mayo I've found in the store is at least part soy. I had a whole lot of sandwiches moistened with Italian dressing (homemade) or mustard until I learned to make my own mayo. I'm way too lazy to retype all these directions so I'll just link over to these precise directions with pictures and add a few tips.
I cannot make this work in my KitchenAid mixer. I don't know if the bowl is too deep for the limited amount of ingredients or what but it doesn't work. It does wonderfully with one beater in my hand mixer. According to the comments on the link it works even better with a stick blender.
Safflower is as close to a flavorless oil as I've ever found. It doesn't leave the mouth feeling as greasy as olive oil but it also doesn't have the health benefits. I usually use 1 part olive to 2 parts safflower.
You cannot cook with this mayo. It will become a big puddle of oil the minute it gets hot.
This is raw eggs. I'd suggest avoiding this recipe if you are pregnant or immune compromised. I don't give it to my kids either. If you've ever dealt with a toddler with gastro-intestinal upset I don't have to explain.