I loved pretending to ride with stick horses
by mylue as a kid which is why I love the idea of relay races on stick horses.
Depending on how expensive you want this party to get, you could buy everyone a stick horse, which on the cheap end is $ 14 per horse, or you could just have everyone bring their own. You could also just get one horse per so many girls and make them take turns using the stick horse.
In terms of the actual relay race, you could have a jumping course with household objects where fastest time wins or you could have a horseshoe that they have to get in a bucket and fastest time there wins. Unfortunately it is more of a summer, spring, fall game.
Cost: Less than $ 15 if you buy only one stick horse