What are some alternatives to xanthan gum for stabilizing mayonnaise?
Xanthan gum emulsifies by being a hydrocolloid, so agar and acacia have a chance of also working since they are also hydrocolloids. The amounts would have to be altered as agar sets much more solidly than xanthan gum, and gum acacia sets less solidly. 

Lecithin is a commonly used emulsifier in mayonnaise (probably even more so than xanthan gum). It's normally obtained from soy, but sunflower lecithin is also available. Usually a fairly small amount is sufficient. My experience is with soy lecithin, but only a very small amount was needed: about a teaspoon to emulsify a quart of mayo.

Of course,t he most traditional emulsifier is simply egg yolks, which most homemade mayonnaise will already contain. Unless you need to keep it emulsified for really long periods of time, the yolks alone should suffice.

Even without the eggs, you can keep mayo emulsified for shorter periods of time using vegetables. Serious Eats did an article on it.