Mayo is made up of egg yolk, oil, and an acid (like vinegar or lemon juice). Some vinegars are chametz or kitniyot, which may not be consumed on Passover. But some are perfectly OK. In addition, there may be other chametz ingredients or traces in your commercially produced vinegar.

As such, use only mayonnaise with kosher-for-Passover certification, or make your own (here is a great recipe!).