I have never, ever known a piercer in the UK to refuse to pierce a teenager with parental consent, not for something ear related anyway! Most say 16 without parental consent, which I think is fair enough as that is the age of consent, and whilst not a full adult, you're allowed more autonomy in law at that age.
WRT Claires, and other chain stores:
-It is better to be pierced with a needle than a gun, due to needles being more sterile (one use only). Piercing cartilage such as the upper ear with a gun is more painful and can cause permenant disfigurements, so should not be done.
-A professional piercer will be able to advise on the best jewellery to promote healing and prevent allergic reactions.
-A piercing studio is more likely to have qualified first aiders, in the rare event something does go wrong. Also, as they have more experience, espeicially with non-earlobe piercings, things are less likely to go wrong.
I think getting first ear piercings in Claires is ok but not ideal, however for second piercings or other ear piercings I would definately go to someone who would use a needle and had more experience and training.
Honestly between now and 16 she is likely to go behind your back and get it done anyway, possibly by a friend. At least if you allow it, you can chose where she has it done, and ensure everything is safe as it can be. Unstretched lobe piercings are very likely to disappear entirely if she takes them out when she's older, if she wants a job where they would be an issue.