Yes, you can absolutely make a wonderful Amaretto Sour without egg white. While egg white adds a luscious, velvety foam and softens the sharper edges of citrus, it is entirely optional, especially for a relaxed evening at home.
Historically, the drink became popular in the 1970s without any egg white at all. Without it, the texture will simply be lighter, crisper, and more refreshing rather than thick and creamy. To achieve the best possible texture without egg white, be sure to shake your ingredients vigorously with plenty of fresh ice for about twelve to fifteen seconds. This chills the drink thoroughly and introduces tiny air bubbles that give the liquid a clean, pleasant lift.
If you find that you miss the dense foam cap but still want to avoid raw eggs, aquafaba (the liquid from a can of chickpeas) is a very reliable alternative. Half an ounce of aquafaba behaves almost identically to egg white in the shaker without imparting any raw egg flavor.
Just keep in mind that egg white usually tames acidity and sweetness slightly. If you are building your drink with amaretto, fresh lemon juice, and a splash of cask-strength bourbon to anchor the sugars, you may want to taste test after shaking and make minor adjustments to your lemon-to-sweetness balance to suit your palate.