Does my Toddler Need Arch Support?

You do not know how many times a day I get this question!
In short, unless your pediatrician or podiatrist has found something wrong with your toddler's foot, no.
Babies are born with flat feet, often covered by a protective layer of fat (although some children can be born with very narrow feet as well). This is healthy and normal! You may notice that most toddler shoes do not appear to have an arch, and this is for a reason.
In most children the arch will not start to develop until they are at least 3 years old. Even at this age, however, arch support is not necessary. The bulk of childhood arch development will take place between the ages of 5-7 with most kids not having a fully developed arch until 8-10 years old. 
As far as the risk for developing flat feet is concerned, only 1 to 2 out of 10 children will actually have flat feet, and the support (or lack thereof) of a toddler shoe will not impact it.