The control itself? Unlikely. They're not made to be taken apart and if the internal parts (like contacts or diaphragm) are worn out those are no more available than the complete unit. What does it do? Not ever shut off? Or not turn on?
But that doesn't doom the fridge at all. First, hang on and see if somebody here knows about substituting antique fridge parts. Second, the cold control could be replaced with a digital sensor and solidstate relay, would take a little design work. Even a timed controller could work, say 15min on and 15min off. It couldn't account for how often the door gets opened but over time that's about what a fridge averages out to, half on, half off.