There is no such thing.
Some piercers claim to be licensed in one way or another. This is at best 50% true. There is no central body qualified or legally able to offer licensing to piercers or tattoo artists. Thus, there is no piercing license.
In some states, the state does license piercers, but this is largely a health board license which ensures that studios adhere to a set of rules governing use of an autoclave, sterile procedures, etc. It does not guarantee in any way that the practitioner is technically qualified to perform piercings or tattoos.