You can purchase the property with your LLC if that LLC exists...therefore you need an operating agreement, you are a member, LLC's do not have owners, or you authorized to sign on behalf of the LLC. But in any case that particular LLC needs to be real.
If you write an offer with an LLC name, negotiate the short sale while you advertise for an end buyer, then get the approval, and then have the end buyer become the LLC member...then technically you wrote an offer with a fake LLC name and would be considered a "straw buyer."
I have seen some people do this successfully...but would advise to not take part in it...I am not an attorney but writing an offer with a fake LLC sounds illegal.
But here is how you can do it legally. Create the LLC first by going through all the normal steps, you are the member, so you can legally write the offer with the LLC as the buyer. Once you get the approval and the end buyer you can then ask the foreclosing lender if they will change the name on the approval letter...this is assuming that the approval letter was specific enough to state that your LLC is the only entity or person that can close this deal. Some approval letters are not specific so anybody can close the deal.
In these cases though you will either be attempting to double close, simultaneous close, or have the end buyer close and pay you an assignment fee or finders fee outside of closing (pending the wording of the approval letter of course)
There are quite a few ways the ABC can work...just make sure you aren't breaking any laws.