Blitzing silken tofu with garlic, salt, lemon, a little cayenne and some oil gives you a creepily good substitute for aioli. On a side note, you can also make a good faux Caesar dressing by adding mustard powder, anchovies, and grated parm to the tofu aioli and thinning it with a little water.

I have not tried it, but unflavored Greek yogurt with garlic might work out nice.

Here's a recipe for a milk-based, eggless mayo from Amanda. You could use this as a base.
http://www.food52.com/recipes/3041_milk_mayonnaise_maionese_de_leite

You could make a garlic paste and whisk it into some Vegenaise (a good quality eggless mayonnaise) then thin it to an aioli like consistency. Vegenaise is available at Whole Foods and health food stores.