Employment Law Questions? Ask an Employment Lawyer.
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You can apply but it's going to be extremely difficult to get it since you have a job waiting for you and the unemployment office is almost certainly going to consider it as quitting as opposed to being terminated.
I've never seen them make an exception based on an inability to find child care. What I would suggest if you want try that is to contact as many child care agencies as you can, get the numbers and names of who you spoke with, so you can support the fact that you were unable to find alternative care and had no choice.
They have no requirements at all past the FMLA date where the 12 weeks expire. They can tell you that day or any time after that.
You might look around some more, there may be other alternatives for a few weeks including some government options but it all depends on your area.
What state are you in?