In an effort to avoid soybean oil, we only use homemade mayonnaise. My recipe is very similar to the one that Kathy in FL posted. I make mine with avocado oil or olive oil (just not extra virgin olive oil, though as the flavor is too strong).
I've made mayo with whole eggs or with just the yolks depending on if I'll be needing the whites for another purpose. If I'm using yolks only, I usually add about a fourth of a cup of water (to four yolks) after adding the oil.
For the acid I usually use lemon juice, but vinegar works just as well.
Once done, I leave the mayonnaise out on the counter for about an hour so the acid has a chance to work.
The most important tip I've learned over the years is to only use room temperature eggs. You're trying to create an emulsification and if your eggs are too cold or too warm (some people like to scald whole eggs in an effort to kill any bacteria), you'll end up with a runny mess that won't emulsify.