Many rec centers have programs that 12 – 15 year olds can take a class for using the weight room. A parent needs to be present usual after the class is completed. If your local rec center/Y/commercial gym doesn’t offer this, you may be able to negotiate a personal trainer teaching gym etiquette and safety to allow your daughter access.
I applaud your desire and action to include your daughter in fitness.
Also check with local community/recreational centers near you. They are great for what you are looking for, and many times have a youth orientation or introduction to go over the fitness area equipment (this is very important at any location you choose). In addition, you will find many other family activities are offered–a great way to spend time with your daughter!
As others have said, every gym has their own policies on that & it’s directly related to their insurance policy. One of the gyms I belong to is a place called Fitness 19 – a smaller chain gym in my neighborhood. I was able to sign my daughter up at age 12 – provided I was with her. I’m not sure at what age she can be in there without me, but I’m not worrying about that for a while! Simply check around – the Y’s are a great place to start. Good luck Sophie.