score:3
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.)