There are actually two times when you can request repairs on a bank-owned (REO) property.
Your purchase agreement (offer) should include any major repairs that you would like the seller (bank) to complete for you before you take possession of the property. If any other significant items come up during the inspection process, you can prepare an addendum requesting additional work to be done. If requesting additional work after your offer has been accepted, be sure to include quotes from at least 2 licensed contractors.
Be aware that the bank has an incentive to complete repairs in order to either make the property sell faster or to preserve the property (i.e. leaking roof or termite repairs). Anything else is not likely to be considered. Also, the bank will not give you cash back to make the repairs yourself - the bank-authorized repairs will need to be completed before the property transfers ownership to you. This means that you may get new carpet, but it will be standard medium-grade neutral-tan carpet instead of whatever you've set your heart on at the local store.
As others have said, use a realtor so all of these requests are handled correctly, or the bank will not consider them.