It the past I have been offered an upgrade while checking in at the airlines desk. These days checking in online, or at one of the automatic booths at the airport can be quite convenient. I am wondering if by using these modalities I am denying myself the chance of an upgrade.
Yes, you can ask for an upgrade while checking in online or checking in at the airport using a self-service machine. That being said, it's always better to request an upgrade well in advance to increase your chances of getting one. You can do that online either when you buy your ticket or later.
That applies at least to all major US airlines.
Here is the AA web form you can use to request the upgrade online:
I may be wrong, but complimentary upgrades require an action by a human being. So, if you want to have a chance, you will have to check in at the counter.
If the complimentary upgrade is because of your status, you would be notified from the podium at the gate.