My wife and I were married in Florida, so I can only tell you about it from that perspective.
In my wedding, we had an out-of-state priest drive down to marry us. Essentially, you just go to the city hall and get a document (I want to say license, but that is not totally correct. It is more like an 'I intend to marry this person' form and you've got 60-90 days to get hitched.). Then the person that marries you (our priest) signs it, as do (I think) you and your spouse and two witnesses.
The document is sent to city hall and they send you a license back.
It is actually very common to be married by a non-clergy or non-municipal official these days.
I have a friend of mine who has presided over three ceremonies for people.
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One word of caution, though. Weddings are extremely stressful events. Everyone in the wedding party (especially parents) are going to spend a lot of time telling you and your spouse what 'needs to happen'. Having a competent official who is used to dealing with over-functioning, demanding family members is a blessing.
Just a thought.
