They'll let you on, but you'll need to go through every single check/verification that they can possibly put you through. That means speaking with the on-duty supervisor who will have to make a call to someone else up the totem pole who will ask you various questions found on your credit report (what street do you live on? how long? what's the cross street? name a relative. where do they currently live? where did you live 5 years ago? etc. This will take extra time, and they may screen you extra thorough since "I forgot my ID" isn't a very strong excuse for not having it. It just sounds suspicious.
(Last year I had my wallet stolen in Vegas and flew back to NJ with ZERO identifying documents.)
PS: Might want to call the airline to advise them of your situation so they can prepare for you ahead of time at the ticketing kiosk.