Can you be married in a carriage?

Yes, of course. Be sure to tip the driver. He can take pictures. But you need to have an address for the actual ceremony, so the driver will have to stop on a specific location to sign the license (it's less bumpier that way, anyway). Personally, I think carriage rides are better for proposals, and post-nuptial rides. Try the Shakespeare Garden or the Lake or the Harlem Meer or one of the little shelters for a wedding -- or the gondola.