Is chlorine needed for a children's inflatable pool?

No, it's not necessary. Chlorine is used to clean it and when you apply it, it stays for a while. Depending on how often you use pool.

Since it's inflatable -

if you use new water - yes put chlorine everyday to kill bacteria

if you use old water - keep you're pool covered otherwise putting chlorine is useless

basically, you don't need it everyday depending on your situation.