I'm working on a system that will generate electrical power from a mechanical input. If the mechanical input is unable to generate the required torque to meet the rated torque of the DC machine, will the shaft still be able to rotate and therefore generate electrical power?
Yes. Generally an electric machine (motor or generator) has current proportional to torque and voltage proportional to speed. This proportionality goes both ways. If the current demand is lower than rated, the torque can also be lower than rated.
I daresay no electric machine has ever been destroyed by using it too lightly. If you're below the rating, all that means is you bought a bigger machine than you needed.
(They can even be surprisingly tolerant to severe overloads, too, if you have ever seen someone crank their 24-volt motorized wheelchair up to 120 volts for YouTube clicks)