I would start her with a 223 or 222 in a Remington Model 7 youth.
Ammo is no problem and the youth model will fit her. Also as she grows you can get a standard stock that will fit better when she is older.
J E CUSTOM
A Model 7 would work just fine. A Ruger American Compact might be a less expensive option. Choose .223 if you already have or can readily get a supply of small rifle primers. Choose 22-250 if you have or can more easily obtain large rifle primers.
Both choices will easily allow you to load rimfire equivalent ammo using Trail Boss or shotgun powder (800-X, Blue Dot, etc.) and bullets like these:
Quality hand-cast bullets for reloaders, competitors, and recreational shooters - 224 cal Rifle Bullets - Price List
Note that the mold part numbers are included in the information provided, so if you already cast your own bullets, you could save quite a bit of money. Loaded like this, brass life will be VERY long and rimfire level practice ammo will be far less expensive than anything you can get off the shelf.
You also retain the option of loading full power ammo as your daughter's skills progress. That is something no rimfire can offer, even if ammo were available at reasonable cost.
James Calhoon's website has two informative articles on the subject, under the title TO BEE OR 223. Though he is using different bullets than those I referenced above, his information gives an idea of what is possible. The basic concept can be adapted to other .22 centerfires, too. I have done the same with 22-250 alongside a friend's .22 Hornet and .223. All of them worked very well. James Calhoon's website is here:
http://www.jamescalhoon.com/
I have some very nice .22 rimfires, including a CZ 452 Lux, that are well on their way to becoming wall hangers because of the current rimfire ammo situation. My shooting skills, likewise, are deteriorating because I am not getting the reps I need to maintain good fundamentals. I intend to remedy that with a budget centerfire rifle that can fill multiple roles. That may just end up being the final nail in the rimfire coffin for me.