Set the bowl on a damp kitchen towel (to hold the bowl steady) and begin whisking in the olive oil a few drops at a time. Slowly add 1/2 cup of the olive oil, whisking constantly; as the aioli begins to thicken, you can add the oil in a thin stream. When the aioli is very thick, add 2 teaspoons of the lemon juice. Gradually whisk in another 1/4 cup of the olive oil, then 2 more teaspoons of the lemon juice. Whisk in the remaining 1/4 cup of olive oil, add the 2 remaining teaspoons of lemon juice, then whisk in the vegetable oil. When all of the vegetable oil has been incorporated, season the aioli with lemon juice and sea salt to taste.