Sorry but if my child of 12 wanted to become vegan I'd be making it clear that they'd have to do all their own cooking from now on. That should cure it pretty sharpish.
I don't think many 12 year olds are capable of understanding what they need to eat as vegans in order to be properly healthy, or of quite how limiting and puritanical it is going to be for them in the long haul. You have a family to cook for which is tiresome enough without having to start with this nonsense.
If she wanted to go vegetarian while the rest of you remain meat eaters, then that I would cooperate with. a) because it's easier and b) because it's much easier to make it a healthy and nutrionally complete than veganism.
I simply do not believe that veganism is a sensible or healthy way to eat, especially not for a growing child about to go through puberty. While she is still a child you have to be able to put your foot down over certain things if you do not believe that they are in her best interests, just as you would with anything else.
I'm not suggesting you force feed her food she doesn't want to eat, but I would be telling her that if she wants a vegan diet she will have plan it, shop for it (you pay obviously) and to cook it for herself and that you will be happy to help with vegetarian food but nothing more than that. And insist that you check her menu plans to make sure they are not massively deficient nutritionally.
I think perhaps she ate the food at your MILs because she didn't feel robust enough to have a discussion with her about it and did it for an easy life. Whereas with you she feels more confident about making demands at home with you and DH.
I agree with your DH, don't pander too much to her and do keep an eye out for signs of an ED. Quite honestly if you don't bend over backwards to facilitate this I imagine it will fizzle out pretty quickly. Especially once she actually tastes most vegan food.
The very real danger is that she will want to start living on sweets and crap like crisps because it's easier and less challenging than eating proper vegan food and she's hungry all the time.