"In my experience, when children make things up about others it's often to test the water and see what others think because, sometimes it's about the person themself having done the wrong deed. Eg. A girl tells mum that her 'friend' sent an explicit text - see mums reaction to gauge if she should fess up because it's actually her that sent it. Or a boy tells a friend that 'x pee'd on the floor' to test the reaction because actually it was them."
^^This, from @NameChange245 sums up my thoughts, more precisely.
@Becra , whilst I see why you went to the shops in question, I do agree with others that it was probably a futile exercise. If they'd seen your daughter steal - which is essentially what you were asking of them, right? - then they would have held her until you and/or the police arrived. Shoplifters get prosecuted these days, as a matter of course, regardless of their age. So even if your daughter did steal something from them (and I'm not suggesting that she did), then they would have been oblivious to it... but if they remember her face, and they very well might do, then she might find shopping there again results in her being very closely watched. Because, well, no smoke without fire...
I also doubt you'll get an apology. Because the dubious friend has obviously doubled down, despite the written statements from the friends she said "oh, Becra's daughter stole..." to possibly proving that she said this. Although wouldn't the school have told her mother what the allegations against her daughter were, what the written statements said about her, before releasing her, upset or not?
The friend's saying that she didn't lie, and her mother believes her.
Your daughter's saying that she didn't steal, and you believe her.
Every mother wants to believe their own child. It's hard-wired into us to want to protect them, no matter what. But... and I do mean this kindly... you and the other mother have dug yourselves into defensive trenches that now mean this is A Situation which your daughters are the ones who have to navigate their way through, regardless of whether they remain friends or not. Your daughter is going to lose friends over this, just as the other girl is. Factions will start to form, with some friends believing your daughter, and others taking the side of the friend.
Leave it to the school to deal with, but do keep a weather eye nonetheless. Because this has already escalated to communications between parents outside of school, which means that they may well wash their hands of sanctions/consequences for either girl (it happens - one of my son's friends did something at school, the parents took to social media, the school excluded him permanently as a result of that... but his something was vastly more extreme than shoplifting/lying, to be fair).
Keep your eye on the situation, be on top of it if/when it escalates to accusations of bullying (which I suspect it might, actually), but let the teachers sort it all out. They're the unbiased experts, after all.
But do send your daughter in tomorrow. If she's off, then she'll be found guilty by the court of her peers. She needs to go in with her head held high, and yes; you need to go in with her and insist on a meeting with her head of year, so that you can tell them your side of this awful situation and ask them to keep an eye on it (in loco parentis, and all that...). It's not easy being 12, and it's equally not easy parenting 12 year olds. But you do need to trust your daughter, and the school, enough to take a slight step backwards. Ignore the other mother completely. She's just doing what you are - believing her daughter, blindly and wholeheartedly. It's called being a parent.