Well, this has been covered about a dozen times, but three or four more times won't kill us, right?
The 5.56mm cartridge is a NATO standard shell, and yes, it's available to the public.
The .223 is the civilian version of the same round, but at a lower pressure.
A 5.56mm chamber will accept a .223 round and it's regarded as "safe". The dimensions of the cartridge dictate that a 5.56mm chamber firing a .223 round won't be as accurate as a 5.56mm round out of a 5.56mm chamber.
It is generally regarded that if you put a 5.56mm round in a .223 cut chamber, you COULD, POSSIBLY
, generate enough pressure to cause a KaBoom.
If you put a 5.56mm round in a .223 chamber, it won't be as accurate as a .223 Civilian round fired from a .223 chamber.
I have never, PERSONALLY, seen a problem caused by putting a 5.56mm round in a .223 chamber - but the Interwebz is FULL of stories of it.
For your own protection, if you EVER think you will get your hands on 5.56mm ammo - then get a 5.56mm chamber.
For the record, once the Space Zombies start wandering the Earth, you will find a TON more 5.56mm ammo in the after world, than you will .223