Where can I get a new phone that will ship to my APO where I am deployed?

If you have an APO Amazon will ship to it.

For anything that Amazon or anyone else cannot or will not ship to an APO/FPO, you can use APO Box. I've used this service on both deployments and while assigned to Korea. Basically, for a small fee, they give you a US Stateside address where your item(s) is/are shipped. They then forward your stuff on to your actual APO/FPO address using either Military or other cargo carriers that deliver to your location. Again, I have used this service, and it works reliably.

http://www.apobox.com/

If you have civilian contractor buddies they are a great resource. We used to hand-carry items back for our green brothers/sisters and most times it was faster (def safer) than shipping.