If you miss your connection, and you are traveling on one ticket, then the carrier is responsible for putting you on their next available flight. Note that this does not mean the very next flight, it means the next flight with an empty seat. Possibly that that will not be until the next day. They will not provide food or accommodation in the mean time, or any other sort of compensation. Also, since you never know whether there will be a seat on a flight or not until it is about to depart, you will have to wait at the airport.

The first thing to find out is how many more ORD-LGA flights there are on this particular airline after yours that day. If there are at least 1-2, then I would say go for it if the fare is good.