She’s the clerk. I suspect her name is on the form.
And? If she's not the one issuing the license, or taking any part in the issuance or processing of the form, how does the fact that her name is on the form (as a technicaly matter, in her official capacity as clerk) violate her religious freedom in any way?
That is exactly the point, she said she didn’t want her name legitimizing something that was against her faith and that furthermore was not written into state law that she took an oath to.