If you're allergic to eggs, or just don't eat them for whatever reason, you may feel left out from a few food categories, but creamy aioli doesn't need to be one of them. There are actually two tasty recipes that are egg-abstainer friendly, and both are super easy to make.
The first is just a simple emulsion of olive oil and garlic, along with a little lemon juice and salt. Watch the above video for the full procedure, but the first thing you're going to need to do is mash up your garlic really well with a healthy sprinkling of salt using a mortar and pestle. Once you have a nice paste, drizzle a teaspoon or two of lemon juice in there and work that in. Next, gradually mix in a 1/4 cup of light olive oil, drizzling in a little bit at a time and mixing each aliquot in with the pestle until creamy smooth.
The second recipe requires a blender, but is almost no work at all. To make a creamy, rich mayo-like sauce that also happens to be vegan, you'll just need a handful of slivered almonds (1/2 cup), a 1/4 cup of olive oil, 1/2 a cup of water, some salt and a blender. Just chuck everything in there, turn the blender on, and let it do its thing for three minutes. When it's done, you'll have a tasty sauce that can be used just as you would mayo in sandwiches, salads, and spreads.
Real Aioli for Real [Food Wishes]
Almond Aioli [Bon Appetit]