Do goldfish need an air pump in the tank?

Short answer: No. But it depends on how much water they have.

Goldfish can be kept in ponds for example. The larger the surface area of the water in your aquarium, the more gas exchange will occur, and the easier it will be to maintain proper amounts of oxygen in the water.

The point of an air pump is to agitate the surface of the water. This encourages gas exchange at the surface, which is what puts new oxygen into the water. Contrary to popular belief, the bubbles themselves do NOT add oxygen to the water. It is the surface agitation they cause that aids in oxygenating the water. Gas exchange will happen even in completely still water, but surface agitation enhances it.

In an aquarium, you probably need some kind of surface agitation unless you have a very large tank. Power filters can themselves provide adequate surface agitation, but having an air pump is never a bad thing.