You certainly can, but it may be more difficult. I would start by trying the other methods outlined in this article. If those don't work, then there is likely something mechanically wrong with the vehicle.
It can be as simple as replacing an easy to reach $10 sensor that has gone bad, or much more difficult. While some expert mechanics can troubleshoot their vehicle and locate the issue without the assistance of an OBD reader, most people (such as myself) need a code reader to make things easier for us.
A code reader can help you by manually allowing you to clear the fault code that has caused the check engine light to illuminate (in case it was stuck on due to an error in the car's computer), or if there truly is something mechanically wrong then it can give you the "error code" that your car is producing.
This error code is the actual reason the check engine light has been activated. You can then look up this error code in the reader's manual or online and see what the issue is with your vehicle in plain English. It is a handy tool that even most expert mechanics quite heavily rely on these days.