Basically, one of the requirements to operate a Chicago cab* on any given day is to be able to accept credit cards. Because of this, if the driver flat out refuses to take your credit card they're not eligible for the fare. If you're really having problems with a driver I suggest calling 311 (or threatening to at least). I've never had to do this, but when I called to formally complain about a taxi the first question they asked is "are you in the cab?" so they must have some sort of action plan for that.

*I believe this only applies to fleet cabs (not independents), but that's virtually all of them anyway.

Note: 2am so I'm not going to look up the specifics right now, but if you want to know more, feel free to email me (in my profile).