Can you get complementary upgrades when you check-in online or through a booth at the airport?

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.

American Airlines

Request an upgrade when making your flight reservations on AA.com, through an American Airlines agent, or your travel agent. [...]

If your upgrade hasn't been confirmed as of the time of check-in, you will automatically be given the opportunity to be added to the Airport Upgrade Standby List while checking in via AA.com, your mobile device, Self-Service Check-in, or with an airport agent.

Delta Airlines (see also)

When you make your reservation and provide your SkyMiles # at delta.com, over the phone, at the airport, or through a travel agent, members will receive an automatic complimentary upgrade request. [...]

If your upgrade is not confirmed at the time of check-in, you can be added to the upgrade standby list at the airport. This can be requested at a kiosk or through a Delta representative. Be sure to add yourself to the list no later than 30 minutes before departure to be eligible for an upgrade.

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.

There's nothing stopping you from asking at the check-in desk or at the gate, even if you have already checked-in online. However, airlines have become much more formal about upgrades, so these days it is quite rare to get an upgrade "just by asking."

I have been given an upgrade just once.
Flying with Iceland Air, Europe to Canada with a 10 day stop on Iceland on the way out, somehow I had a class better than cattle class which allowed the long stay within the one ticket, not much more expensive.
Having checked in online I arrived at the airport not expecting anything but just walking onto the plane. My name was called to come forward at the gate, was given a different seat number and ended up in the frontmost part of the plane.

I was given an upgrade while I did not ask for it, checked in online and not a member of any regular passenger scheme, it was even the first flight with the company.
Nice seat and good food, but not worth the extra money they charge for the class.

So it does happen but going by my experience, not often.