The people of New York State will have to change the law. But, until then the clerks will be forced to comply with the law, just as they were forced to comply with the law prior to the legalization of gay marriage.
Religious freedom is constitutional--meaning it trumps all other laws. You cannot say, you MUST OBEY, until the law is changed--when it comes to longstanding religious conviction.
The government has always made reasonable accommodation to persons of faith--over long-standing religious convictions. For example, Mennonites during WWII were not FORCED to fight in the Army--since pacifist non-violence is a long standing essential to their religion. Many Mennonites and other pacifist Anabaptists DID serve in the military during WWII, but in non-combatant positions (Medics, in hospitals, food service, etc)--where they were not required to violate their conscience.
To try to CRUSH a woman's career, who, very discreetly, merely asked for appointments to get marriage licenses--in an office which only does 10 licenses a year, is very definitely the State running roughshod over freedom of conscience--on a VERY controversial issue.
This proves once again--as I've warned in FR forums over and over--homosexual rights and religious freedom are NOT compatible.
“But, until then the clerks will be forced to comply with the law, just as they were forced to comply with the law prior to the legalization of gay marriage.”
Lex mala, lex nulla. No person is under any moral compunction to obey that law, or to submit to punishment for refusing.