Lonnie is correct.
To purchase a "concealable firearm," go to your county sheriff's office, fill out a permit to acquire, and wait about a week (no more than 7 days; it can't cost more than $10). You can, in the meantime, find a gun you like and place a deposit on it; I haven't yet seen a shop that won't do this. When you pick up the permit, just head on over to the gun shop and finish up the NICS paperwork (a completely different thing than a permit to acquire), give them the rest of your money, and take it on home.
Interestingly, there is no requirement to otherwise register firearms. Some police agencies might, if you are involved in a shooting, take the weapon and make you get a PTA before you get it back, but it is not illegal to possess a handgun without a corresponding PTA.
It is a funky and confusing system, and it is completely redundant ever since the federal NICS check was mandated. There is no reason for it to exist anymore.