According to Estes, model rocketry is a legally permitted activity in every state in the US. However, four states have laws and regulations that specify certain age or permit requirements. It could be that the person lives in a state such as California, New Jersey, North Dakota or Rhode Island which have age and other restrictions on model rocket motors. None of these states require a permit, but the local Authority Having Jurisdiction might issue a permit to launch a rocket, which could be interpreted as a rocket license.
Since the question is, Is there such a thing as a “rocket license” in the US? I can answer that question, at least in terms of the state of California. CalFire - The Office of the State Fire Marshal is the state agency that handles licensing requirements for rockets, pyrotechnics and special effects. As a licensed pyrotechnician, I use the same forms and handbook as the rocket operators use, I figured I would share the details on licensing in California. You have to be 18 years of age and pass a background check for the pyrotechnic licenses, which is a Federal requirement, but I did not notice a similar age restriction for a rocket license, but I will assume one exists.
Pyrotechnic Operator--Rockets First Class may conduct and is restricted to all activities in connection with research experiments, production, transportation, fuel loading and launching of all types of experimental rockets. Such operator shall also be responsible for the actions and conduct of all assistants. Operators licensed under subsection (b) are also required to obtain a local permit from the authority having jurisdiction prior to all launches.
Pyrotechnic Operator--Rockets Second Class may conduct and is restricted to all activities in connection with research experiments, production, transportation, fuel loading and launching of all types of solid fuel experimental rockets only. Such operator shall also be responsible for the actions and conduct of all assistants. Operators licensed under subsection (b) are also required to obtain a local permit from the authority having jurisdiction prior to all launches.
Pyrotechnic Operator--Rockets Third Class may purchase, transport, store, and launch high power rockets. Experimental high power rocket motors may only be imported, exported, and wholesaled by individuals or companies holding valid import, export, or wholesale licenses. Pyrotechnic Operators--Third Class may only purchase high powered rocket motors from licensed wholesalers. Operators licensed under subsection (b) are also required to obtain a local permit from the authority having jurisdiction prior to all launches.
I am not clear if a rocket launched at a site like Vandenberg Air Force Base requires a First Class license, but what I interpret is that if you have that license, you can launch anything, although the specific language does state "experimental".
Once you are licensed by the State of California, you get a credit card sized identity card with your photo, your license number and expiration date. Every time you renew your license, you get a reflective sticker identical to the shape and material the state uses for licensing motor vehicles, although usually a different color.
For specifics for licensing in other states or countries, I suggest looking over the Wikipedia link below.
If @KASHIKAA S ever determines if a license is available, I hope someone posts the information. I think it would be great if minors with an interest in rockets are able to pursue their dreams without age-related restrictions.
Good luck.