My experience with kids is that is more a matter of "courage" than age. Some kids are simply less gun shy than others. Having said that the No.1 mistake that I have seen made introducing kids to recoil is the poor fit of the rifle or shot gun.
A 20 ga. kicks pretty good when you compare it to some of the more popular center fire "deer rifles". So, first thing I would do is fit the shotgun to the kid. Which is a tough call right now because a 9 year old is likely to hit growth ppurts, and be 2" taller in 12-18 months down ther road!!!
Try this. Look at an H&R Untra Slug "Compact". This is a rifled barrel single shot that comes in 20 ga. and will not break the bank. It is more or less a "youth size" ... which means that the "Length of Pull" will be about 13" from the factory. Probably too long for here now. Order a spare stock and put it aside. Take the kid to a gunmith and have one of the stocks fitted to her now. If need be, you can have "shims" added later as she grows. By the time she in in her teens, she'll probably be ready for the full size stock.
This H&R does not have sights, but I think comes with a scope mounting rail already attached. I could be wrong. Put the best scope on it that you can justify right now ... a compact size in something like a 2-7x32 mm or 40 mm would seem to me to be the ticket. If she gets into hunting, then I'd suggest to upgrade to a very good scope and she'll have a 0-150 yards deer killing set-up good for years to come.
Good luck and hope she gets a big one !