Why do egg dye recipes include vinegar? I have seen recipes for easter egg dyes that call for 1/4 cup vinegar per cup of water and others that call for 2 teaspoons per cup of water. That's a pretty wide range--what practical effect does the amount of vinegar have?

More generally, what is the role of vinegar in dye recipes? What would happen differently if it were left out?
Long story short: It's the acid, and any acid will do.

Food colourings are a type of dye known as an acid dye. Their effectiveness relies on hydrogen bonding which only works in an acidic environment.

It's not that the vinegar does anything special - not exactly. Rather, adding a few drops of food colouring to a large amount of water would give you something that's not acidic at all, which will diminish or totally eliminate its effectiveness as a dye. Including vinegar in the mix simply allows you to stretch your rations, so to speak - to dilute the colouring in a whole lot of liquid without making it useless as a dye in the process.

Basically, you're creating a dyebath, which is a well-known term in the textile world, as it's used to dye wool and other fabrics. Warm temperatures (140-180° F) also aid in dye absorption, and apparently - although I'm no expert in textiles - a small amount of Urea also helps.

You could definitely use cream of tartar, citric acid or any other acid, but the recipes using vinegar are generally aiming for a specific pH, so you'll want to adjust quantities of any other acidifier to match. Presumably, vinegar is just the cheapest and most widely available acidifier.