You guys are acting as if she is some kind of martyr, which is ridiculous. No one is forcing her to work for an institutions that does something she disagrees with. She has no real principles, she wants to have her cake and eat it too. She wants their money, but refuse to do the job she is assigned to do. Not to mention that she is not exactly a beacon of the sanctity of marriage (as if that is a factor in marriage in a purely secular form anyway) herself, last time I checked she has married 4 times, 3 divorces.
Let me rephrase the last thing I wrote as a question:
Would you support her if she was a conservative Muslim who refused to issue a marriage certificate to a non-Muslim man and a Muslim woman?
If not, you are simply a hypocrite. The US is not a Christian theocracy, appealing to church laws or biblical principles is irrelevant. The only thing that matters for a person who wants to earn their living off of a government institution is to carry it out according to the laws of the secular government.