Hellow mhorgan,
Interesting question. I was asked this by a lady at work, it a church would marry someone without a license from the state. Curious, since being married to me was always something that I ascribed to as one whose domain was God's and not man. (Matthew 19:6) but have no problem acquiescing to the state requirement.
I asked my pastor this very question and still have not received an answer. He's contacted the Headquarters and they have a lawyer poring over documents to determine the churches stance. My Pastor stated, that according to scripture that a license was needed. I told him I could not find that scripturally, but only found a license for a divorce, which brother B-A-C has quoted. James1523 brought up another good point as well about submission to authorities, but as always, the limits to that authority and adhering to scripture can not always be reconciled, but this is not the place for that discussion.
I do like the comment that Fragrant Grace made in regards to her own marriage. For if Christ is not the head of the marriage and in each of your individual lives which are to be joined. Whose providing the guidance and help when the river gets hard to navigate?
I assume you and the man you intend to marry attend a church. Ask the Pastor the same question. Will you marry us without a state license, so that we may enter into this blessed covenant between us and God, and do so by a Shepherd of the Lord.
The only problem I see is that you haven't stated why you don't want to get a license from the state. Because from what you've written it seems it's a personal decision and not a scriptural one?
Praying that it will as always, provide a joining of two believers in Christ Jesus, who become one to the Glory of God.
YBIC
C4E