Crush saltines so that they resemble coarse meal, not dust (about 1 1/2 cups crumbs). Stir together crushed crackers, melted butter, sugar, and egg white in a medium bowl until combined.
Transfer mixture to a 9-inch glass or ceramic standard pie plate greased with cooking spray; firmly press on bottom and up the sides. Freeze for 10 minutes.
While crust cools, whisk together condensed milk and egg yolks until smooth. zest 1 of the lemons and 1 of the limes to yield 1 teaspoon each. Juice zested lemon and 2 of the limes to yield 1/4 cup juice each. Whisk lime juice, lemon juice, and 1 teaspoon of the vanilla into condensed milk mixture until smooth. Pour mixture into warm crust.
Preheat oven to 350 degrees F (175 degrees C).
Bake in the preheated oven until crust is lightly browned, 15 to 20 minutes. Transfer to a wire rack; cool slightly, about 10 minutes.