Unfortunately there are gads of neighborhoods throughout the whitetails' range that are overpopulated. Residents go from liking to see an occasional deer to some of the residents hating deer. It is sad when deer are left unmanaged and then become hated. They are beautiful creatures and deserve respect and to be appropriately managed. I'm not aware of any group that will move deer. Deer are owned by the state (unless part of a captive herd) and can only be moved, harvested, etc., by special permit issued by the governing state agency. Problems similar to yours have been faced for years and the only solution to date is to harvest the excess deer. There are several major metropolitan areas that accomplish this by hosting archery only hunts. These hunts are strongly regulated, and often the hunters must qualify by taking a safety and proficiency test. If you and your neighbors wish to address the problem I'm very confident this is the only route. You may wish to contact the regional Missouri Department of Conservation office and seek addition counsel since this would be a state/local jurisdiction sanctioned event.
Whatever you do, don't allow the neighborhood to lose respect for a beautiful part of God's creation. The deer are not at fault.
Growing Deer together,