new imac power adapter

Did you purchase this iMac directly from Apple or an Apple Authorized Reseller? ... or from another third-party source? I ask, because your answer will determine what options you may have to get this resolved.

Typically, if you did purchase directly from Apple or an AAR, the Mac's power adapter would come with the same one-year limited warranty against manufacturing defects. Should that be the case, you should be able to get a replacement adapter at no cost to you. How soon you can get that adapter would depend on if you go to a physical or online store.

Can you lookup status of this case anywhere? For example, the Apple Support App? Or are you left with no option but to deal with mere humans?

Hmm, then I'm surprised that they didn't offer you a replacement then and there. AFAIK, they don't repair them.

If they didn't have a replacement available, I am also not aware of these adapters being in short supply that would require more than a few days to restock the store.

Do you have any paperwork or electronic record of this interaction with Apple Store?