Okay, the end to the gun saga. Or at least, an intermediate stopping place (there's still the ten-day wait).
Hey, I own a gun. I just bought a Remington 572 22LR. The LR ammunition is supposed to be the cheapest of the .22s (it's running about a dime a round, which is fine). It was used, but I liked the pump action better than the lever action of the new Ruger 22 they had. (You don't have to drop the gun to pump the next round into the chamber.) The sight lines up well for me, and it fits nicely into my shoulder. It's light enough that I can hold it out and take my time aiming. I also bought a used suede/sheepskin case, which the guy said was a good choice because it's breathable. Anyway, by the time I was done--LR 22 ammo, 22 stinger ammo, background check, safety test/certification, case, cable lock for the case, and gun--I ended up at 474 bucks, so it even came in (barely) under budget.
So, thanks everyone for the help.
Decent gun, but unlikely to reliably stop a coyote. I would have said a lever action .357/.38. Light recoil, easy to use.
In CA legal guns, I believe you can get some KelTecs in 5.56mm.
You can also go with a 20 gauge shotgun--easy and cheap to get ammo for, plenty of power for your needs, and won't have extreme range, because you need to be aware where any missed rounds are impacting with a rifle.
Do please check that it is legal to shoot pests where you are, and if there are any state/county restrictions.
As far as finding ammo, there are other sites like this that will help compare cost and availability. www.AmmoSeek.com