Dept of vital records. Within 7 days of the birth you don't need ANYTHING. You just show up (not sure if you need your baby present or not... we didn't know about this at the time) and declare you've had a baby, get your paperwork done and leave. We did it the "hard" way, but only by comparison. We showed up about 2 months after the birth with my "proof" from the chiropractor, filled out a bunch of paper work and left. It was painless. The only thing that was difficult was having to explain repeatedly that there was no midwife, doctor, etc..
Here's the link for Maricopa County (Phoenix area). The rest of AZ is pretty much the same minus the interview. I got a birth certificate for one of mine when I lived in Yuma and it was as simple as picking up the packet, filling it out, and dropping it back off. I've had clients get them here around Phoenix and they say the interview is a pain and can be picky about what they will accept for proofs.