I just saw that Covid-19 testing is available at ORD in the Bus/Shuttle Center in the main parking garage (link). The website says it is open 06:00 to 18:00, 7 days a week.

Sometime this month they plan on opening a drive-through test center in Parking Lot H.

No walk-ins, a prior appointment is required. And you need to show proof that you require testing, either because of travel or airport employment.

If this works out, I'll try it on my upcoming South Africa trip in a few months.