Rose Marie Belforti is a 57-year-old cheese maker, the elected town clerk in this sprawling Finger Lakes farming community and a self-described Bible-believing Christian. She believes that God has condemned homosexuality as a sin, so she does not want to sign same-sex marriage licenses; instead, she has arranged for a deputy to issue all marriage licenses by appointment. But when a lesbian couple who own a farm near here showed up at the town hall last month, the women said they were unwilling to wait. Now Ms. Belforti is at the heart of an emerging test case, as national advocacy...