I am not sure what you mean by "wire", SMAW is a "stick" rod and we usually do not refer to it as "wire". Anyway to answer your question, you will only need a storage oven if you are using low-hydrogen electrodes (i.e. E7018, E7015, E7016). From a practical standpoint I have found it extremely difficult to consume a 50 pound box of rods within the allowable exposure time limits, and as your tensile strength goes up your allowable time goes down (i.e. E11018 has only a 30 minute limit).
The other problem is that once you exceed the exposure time limit, low-hydrogen electrode can only be rebaked once. AND the holding oven usually does not operate at high enough of a temperature to rebake the rods.
The next problem is after they have been rebaked, how do you identify which rods have been rebaked and therefore cannot be rebaked again.
I would suggest you think through your filler metal rod storage and usuage. There are many issues that have to be considered and some have large economic impacts.
AND finally, YES the AISC will be looking for storage oven if you are using low-hydrogen electrodes.

You are speaking about two different processes. First -SMAW (shielded metal arc welding) does not use flux cored and hard wire. SMAW uses a covered electrode. The reference to flux cored and hard wire leads me to beleive you are actually using a GMAW (gas metal arc welding) process. In GMAW, most codes do not require a "rod oven". Cleanliness of the material may be assured by covering the spools when not in use. Additionally, Flux cored wire can be susceptable to moisture contamination due to the seam. Care should be taken to not expose the spool of wire to high moisture conditions.