I have an international flight leaving early Tuesday morning. The visa agency has had my passport and application for more than 3 weeks, but due to problems with the photograph and embassy closures they will not be able to get my visa until Monday afternoon. The agency has offered to same-day deliver the visa as soon as they pick it up from the embassy. It's possible that if they can get my passport to FedEx or UPS by 3PM Monday, that it will arrive in the evening by 9PM. But should I call the airline and prepare for the possibility that my visa won't arrive in time? Or is it possible to arrange my visa to be sent to the layover airport (ORD) on Tuesday?
It's up to you, not the airline, to have a valid visa for any countries you may be visiting.
You've not clarified which countries the missing visa is for, but a responsibility the airline DOES have is to ensure you have completed the required visa acquisition for any countries that they fly you to. As a result, if your missing visa is one of these, they're unlikely to want to risk flying you to said countries, even if you promise you'll get the visa at the layover airport (Chicago ORD).
It's generally pretty risky going to any country without a visa/onwards visa - and the US especially don't take kindly to even forgetting to fill out an ESTA for those who don't require visas :/
You COULD call the airline and see if they'll change your flight to give you more time to obtain said visa, but other than that, the airline is out of the equation here - it's between you and the visa company to get it sorted beforehand.
(I realise you've already had this flight and the visa did manage to arrive in time, but I wanted to clarify for future readers.)