for the record, the lawyers for the people trying to get married said they do not want her thrown in jail
They just want her to do her job, which she refuses to do. ACLU has filed contempt charges against her.
Generally, contempt of court is something either the court or a party with standing has to file. Also, there are two different types of contempt: civil and criminal. This, to me, appears more civil. In a civil contempt the judge could hold her in jail until she agrees to comply with the court order. Criminal contempt usually carries a higher burden of proof before a jury and there it must be shown that she actually violated an order and only then could she get, say, x number of days in jail for sure.