How to retrofit existing kitchen exhaust fan
That whole fan assembly will need to be replaced unless you want to do some work hand-making a bracket that will accept a modern exhaust fan motor.  Even then, the fan motors and blades for a modern install probably wouldn't move air efficiently in that size of enclosure.

Your main concern will be finding a new fan that is easy to connect to your 3x10 vent.  They are probably available, but most ceiling exhaust fans that I have seen are made for a round rigid or flex duct.