The Savage Axis/Axis II does not have a 5.56 chamber, it is specifically .223 rem. That being said, most people with bolt action .223's can shoot 5.56 with little trouble, well less trouble than with .223 rem AR's anyway. Shoot at your own risk. If you experience increased effort lifting the bolt or cratered/flattened primers, it's probably not a good idea to keep shooting 5.56 in it.
I have an Axis stainless in .223 rem, and it does have a lengthy throat, which could possibly mean it would shoot 5.56 alright, but I've never tried it. If it were really a Wylde chamber I would think it would say ".223 Wylde" on the barrel. My handloads are with a 69 gr SMK at an OAL of 2.400" and that's 0.010" off the lands.
My guess is most will, some won't safely. There's enough .223 rem floating around in the world you shouldn't need to worry about having to feed it 5.56