Does a dishwasher or garbage disposal have to hard wired into a electric box or can they simply plug into a receptacle?

It depends.
If the manufacturer allows it as a method of installation it is OK. Failure to do what they instruct can void the warranty. In reality, I have seen more with plugs than hard wired, in both applications.

Both can be wired via cord and plug.
Most disposals are wired this way, and more and more DWs are coming with cord and plug from the factory.