I think we should. Hear me out:
See, I'm perfectly fine with churches refusing to perform gay weddings. But a marriage is recognition by a state license. Anyone who is authorized to issue a state license should be required to do so for anyone who legally applies for it, regardless of religious or personal beliefs. You don't see some podunk DMV being allowed to refuse to issue driver's licenses to gays, Jews, or African Americans do you? It's the same issue as a pharmacist being allowed to refuse dispensing legally prescribed drugs because he doesn't think they are being used in accordance with his particular moral view. With authority comes responsibility; acting as a government entity should require acting for all of the citizens of that government.
Otherwise, go ahead and have the weddings and let the couple have their license issued and signed by the county office.