I don't know about the others here but,
I have been through this with my daughter who is now 17. She started shooting at the age of 5 and by the age of nine was sporting two 32 caliber Rugers on her hips, using a a 12 guage 97 and, shooting a 357 Marlin rifle, all quite well I might add. Along the way, somewhere around the age of 6 or 7, I had the same idea as you, at least form the bench she could handle a 223 or a 222 remington. If you can find a remington Model 7 with the youth stock in either of these calibers, buy it. Another gun that may work and will be equally as hard to find, is a Mark X action that was imported by several companies through the years, I believe EAA was one of them. They are a small Mauser action, chambered for the 223 or the 222. The ones that I have seen have all been great shooters. Stay away from the ARs for now. She really isn't ready for that yet. I reecently saw a very compact Ruger 77 with a youth stock that would fill the bill if they chamber it in 223, 222 or 204 Ruger, which has not been mentioned yet, but is a great choice in my opinion.
I think Sinixstar has made the best suggestion so far, of a single shot rifle like the NEF or H&R. These are great shooters at bargain basement prices. The most you will have to do them is clean up the trigger. What you save on the cost of the rifle you can put into a good scope that will always "fit" her. The variety of calibers that these are available in is just short of incredible. I have seen them in 17 HMR, 22 Mag, 223 and 204 Ruger. They probably do have a hornet and even a 222 available, but why bother. The other choice is to build your own single shot, which is the route I went. Back then I was heavy into TC pistols so I simply bought a a youth rifle stock and a 16" 223 barrel for it and Voilla, instant lightweight kids rifle. It was a tack driver too. This is avery good way to go as you can simply buy different barrels for different purposes. As for the caliber, if you are going to use it on coyotes, I would go with the 223 or the 204. I know some guys use the 17 HMR out to a couple hundred yards but, they have probably been shootings dogs longer than your child has been alive. If you reload the 204 is just as cheap to shoot as the 223 and the bullets that are availabe are more explosive than most of the 223s out there. Even if you are shooting factory ammo with with good hunting type bullets the cost of the 204 is comparable.