She can get SSN first using her L2 I-94. L2 is one of the visa classes that can get SSN without an EAD.
https://secure.ssa.gov/apps10/poms.nsf/lnx/0100203500
After you get L1 extension, she can apply for (a)(18) EAD so she will get a longer valid date.
https://secure.ssa.gov/apps10/poms.nsf/lnx/0100203500
After you get L1 extension, she can apply for (a)(18) EAD so she will get a longer valid date.