If you miss your connection from an Amtrak train to an Amtrak train you will ordinarily be put on a later train.
If the next train is the next day, Amtrak will ordinarily pay for a hotel and give you a meal allowance. If you had booked a roomette Amtrak will give you any sleeping accommodation available and not charge you for the upgrade. If no sleeping accommodation is available Amtrak can downgrade you to a coach and refund the difference in fare.
Amtrak does have the right to put you on a plane or a bus but usually does not do that.
This issue is most likely to arise in Chicago. It can happen if you are connecting to a long distance train out of New York Penn Station. If you are connecting to a Northeast Corridor Train there are so many that you would get a train the same day.
Amtrak is usually pretty good at guaranteeing connections if for some odd reason u don't make a connection they will usually just put u on the next train or will compensate u
They may either arrange for you to get on another train or bus. It will depend upon the circumstances of the delay.
Just be patient. Patience is a virtue
Then you wait for the next one.