I've been to ORD when I was silver numerous times, and without a card for many of them, since like the OP, they just don't arrive....

Usually not even asked for it, asked for it a few times, and said "I've lost it", and they looked me up and let me in. Once I found my old, expired silver card (which was still in my bag), thought I had my card after all, and that was a major mistake -- needed about 5 minutes of cross-checks before I was let in.

In other words, you should be fine.

tb

PS if you have Amex plat, you can access AC (but not FL) with that [as a back-up]