What can I do to fix store-bought mayonnaise that split?

Your mayonnaise probably got a little too hot in transit.

Mayonnaise is an emulsion of oil and water. That is, an emulsifier is added to a water-based ingredient that effectively coats droplets of oil and allows those droplets to dissolve into the water instead of coalescing and floating to the surface.

Lecithin is the most common emulsifier used. It is found abundantly in egg yolks.

The emulsion can be damaged if the mayonnaise is heated. When this happens the oil coalesces and you are left with a vinegary base with oil floating on top. Yum.

To fix it you have to reform the emulsion. There are many answers here about how to make or fix homemade mayonnaise. Briefly: You beat an egg yolk well with the water-based portion and then slowly drizzle the oil in while beating like mad until the mayonnaise is thick and white again.

The whole point of making homemade mayo is to use better quality oil and vinegar or incorporate different flavors. It really isn't worth your time trying to save commercial mayo.

Use a stick/immersion blender to give the mix a good mix without adding any other substance to the mayo. It should stabilize this way and I would suggest to keep the mayo in the refrigerator.