If your connection is on the same reservation number, Amtrak will rebook your ongoing reservations itself and also provide a hotel as well a voucher for cash for cab fare and meals. They may provide some kind shuttle bus to the hotel, depends on how many misconnecting passengers they have. You'll probably be advised by an announcement on the train prior to arrival to go to Passenger Services at Chicago Union Station, which will have your revised reservation and hotel. They often used to board Passenger Services representatives down the line and do that onboard,but the last few times I misconnected in Chicago they had stopped doing that. I don't know if Passenger Services is still in the same place, they've been moving things around in CUS lately.
Amtrak will take of the rebooking and you'd go to passenger services even if you were in time for the Lakeshore if you are reserved on the Capitol (which I assume you are.
If the two segments are not on the same reservation, you may well be on your own. I have heard that sometimes Amtrak can put it together and handle you the same way as if you were on the same reservation, but I would not count on that. If they are separate reservations, be prepared that Amtrak said they'd get you to Chicago and they got you to Chicago, and you're on own.
I have misconnected in Chicago more times than I can count. Amtrak is pretty good about taking care of you.