Agree with above posters, get it done at known place. Plus, you'll want to be on top of the cleaning of the piercing, which would be easier at home, but we've had some issues with infected ear piercings on one child, so that colors my viewpoint.
Lots of places to get it done, would recommend "locking" style studs to prevent them coming out (another bit of experience with different child with long hair which inadvertently pulled out the stud, causing a re-piercing experience after the initial holes healed/sealed completely). Most places pierce with a "locking" style, so not hard to find.
Good Luck!