GPs will absolutely consider fitting a coil for a teenager should they choose that option.
Just because they're under 16 doesn't mean they're not allowed access to contraception that adults have.
My DCs aren't teens yet, but part of my last placement at uni was discussing sex, sexual health, and contraception with teenagers.
My advice would be to avoid making her feel ashamed or embarrassed at all costs, if so she'd be unlikely to come to you with any problems she may have in the future. Lay out all her options regarding contraception (and do your research first), show her some contraception websites geared towards teenagers that she browse on her own if that'll make her feel more comfortable.
https://www.nhs.uk/Livewell/Sexandyoungpeople/Pages/Gettingcontraception.aspx
Talk about what methods you used, what you felt worked best, and why. Then ask if she'd like you to make a GP appointment, and if she'd like you to go with her, she might like to talk to the GP on her own, and you should respect that.
Don't try scare her with pictures of diseases or horror stories as some PP suggest
if a teenager wants to have sex, then they will regardless of what happened to some random person they've never met. You sound like you're doing a good job already, I can imagine how frightening this must feel as a parent.