Please go read this:
http://blogs.webmd.com/healthy-skin/2011/09/whats-growing...
The repeated mentions of fungi and bacteria should be enough to convince you that your daughter is not making a safe choice. The mention of "fingers the size of sausages" after swelling due to allergic reactions -- have your kid read THAT.
The fact that so many adult women are posting here that these nails are both expensive and potentially unhealthy is a pretty clear sign that this isn't really about giving a young teen more independence, or letting her spend her own money how she pleases etc. It's an unsafe choice, period, and she is too young to appreciate that. Say no. You will not be popular with her and she won't get it, but it's a time to say that what seems like a simple, fun fashion choice, and a chance to give her some freedom, actually is a very poor health choice.
She's going to complain that she has friends who have these nails, she has never heard of anyone having any problems ever, and so on. Paste on a firm smile and offer her the alternatives below.
I also would bet that she has no idea it's at least $70 a month to maintain these nails. The first time she starts to want something else, you'd have to say, "If you spend X on that you won't be able to maintain your acrylics this month and once you stop maintaining them they'll be wrecked. So, what's your choice?" She'd get tired of having to spend all her money just on maintenance but of course she cannot recognize that right now.
Just say no, on health grounds alone.