Relay Race Tic Tac Toe. You only have 3 items per team (there is never a cats game), so after the 1st three people go, the next people go and move one of their items to a new place. Lots of creative ways to switch this up (create obstacles, 3 legged races, crab walk over to it, etc)
Beach Ball Race- kids work together to hold beach ball between their backs for the first leg of the race, then their sides, then elbows. The field Day Challenge is halfway down on web site & includes water relay, tricycle race, and tug-of-war. Check it out!
Use one long piece of string and string 10-12 donuts through it. Have two adults hold each end of the string and hold it above the kids head. Space out the donuts and have the kids try eating the donuts with their hands behind their backs and using just their mouths.