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You do not need an EAD, to get a Social. To get a social, you need to show that it is required for her to have a social to live in US.
Like, School enrollement, Working, Driving, and tax excempts etc.
In earlier times, School requires a social for a kid to enroll, which can be used to acquire a social. I believe in 1995, they changed the law stating non-immigrants can have a social only if they are employed, or going for a drivers license and for others they provide you with a TIN. I believe lately if you need just a social for only drivers licencse, they only provide you with a TIN. You can apply for a Tin numbr for your kid.